Encoder, decoder, and encoding method

ABSTRACT

An encoder and decoder using LDPC-CC are provided, which avoid lowering the transmission efficiency of information while not deteriorating error correction performance, even at termination; and an associated encoding method is provided. A termination sequence length determining unit determines the sequence length of a transmitted termination sequence added to the end of an information sequence, according to the information length (information size) and encoding rate of the information sequence. A parity calculation unit carries out LDPC-CC coding on the information sequence and the known-information sequence necessary for generating a termination sequence of the determined termination sequence length, and calculates a parity sequence.

TECHNICAL FIELD

The present invention relates to an encoder, decoder and encoding methodusing a low-density parity-check convolutional code (LDPC-CC) supportinga plurality of coding rates.

BACKGROUND ART

In recent years, attention has been attracted to a low-densityparity-check (LDPC) code as an error correction code that provides higherror correction capability with a feasible circuit scale. Because ofits high error correction capability and ease of implementation, an LDPCcode has been adopted in an error correction coding scheme forIEEE802.11n high-speed wireless LAN systems, digital broadcastingsystems, and so forth.

An LDPC code is an error correction code defined by low-density paritycheck matrix H. An LDPC code is a block code having a block length equalto number of columns N of parity check matrix H. For example, Non-PatentLiterature 1, Non-Patent Literature 2, Non-Patent Literature 3 andNon-Patent Literature 4 propose a random LDPC code, array LDPC code andQC-LDPC code (QC: Quasi-Cyclic).

However, a characteristic of many current communication systems is thattransmission information is collectively transmitted per variable-lengthpacket or frame, as in the case of Ethernet (registered trademark). Aproblem with applying an LDPC code, which is a block code, to a systemof this kind is, for example, how to make a fixed-length LDPC code blockcorrespond to a variable-length Ethernet (registered trademark) frame.With IEEE802.11n, the length of a transmission information sequence andan LDPC code block length are adjusted by executing padding processingor puncturing processing on a transmission information sequence, but itis difficult to avoid a change in the coding rate and redundant sequencetransmission due to padding or puncturing.

In contrast to this kind of LDPC code of block code (hereinafterreferred to as “LDPC-BC: Low-Density Parity-Check Block Code”), LDPC-CC(Low-Density Parity-Check Convolutional Code) allowing encoding anddecoding of information sequences of arbitrary length have beeninvestigated (see Non-Patent Literature 1 and Non-Patent Literature 2,for example).

An LDPC-CC is a convolutional code defined by a low-density parity-checkmatrix, and, as an example, parity check matrix H^(T)[0,n] of an LDPC-CCin a coding rate of R=½ (=b/c) is shown in FIG. 1. Here, element h₁^((m))(t) of H^(T)[0,n] has a value of 0 or 1. All elements other thanh₁ ^((m))(t) are 0. M represents the LDPC-CC memory length, and nrepresents the length of an LDPC-CC codeword. As shown in FIG. 1, acharacteristic of an LDPC-CC parity check matrix is that it is aparallelogram-shaped matrix in which 1 is placed only in diagonal termsof the matrix and neighboring elements, and the bottom-left andtop-right elements of the matrix are zero.

An LDPC-CC encoder defined by parity check matrix H^(T)[0,n] when h₁⁽⁰⁾(t)=1 and h₂ ⁽⁰⁾(t)=1 here is represented by FIG. 2. As shown in FIG.2, an LDPC-CC encoder is composed of M+1 shift registers of bit-length cand a modulo 2 adder (exclusive OR calculator). Consequently, acharacteristic of an LDPC-CC encoder is that it can be implemented withextremely simple circuitry in comparison with a circuit that performsgenerator matrix multiplication or an LDPC-BC encoder that performscomputation based on backward (forward) substitution. Also, since theencoder in FIG. 2 is a convolutional code encoder, it is not necessaryto divide an information sequence into fixed-length blocks whenencoding, and an information sequence of any length can be encoded.

CITATION LIST Non-Patent Literature

NPL 1

-   R. G. Gallager, “Low-density parity check codes,” IRE Trans. Inform.    Theory, IT-8, pp-21-28, 1962.    NPL 2-   D. J. C. Mackay, “Good error-correcting codes based on very sparse    matrices,” IEEE Trans. Inform. Theory, vol. 45, no. 2, pp 399-431,    March 1999.    NPL 3-   J. L. Fan, “Array codes as low-density parity-check codes,” proc. of    2nd Int. Symp. on Turbo Codes, pp. 543-546, September 2000.    NPL 4-   M. P. C. Fossorier, “Quasi-cyclic low-density parity-check codes    from circulant permutation matrices,” IEEE Trans. Inform. Theory,    vol. 50, no. 8, pp. 1788-1793, November 2001.    NPL 5-   M. P. C. Fossorier, M. Mihaljevic, and H. Imai, “Reduced complexity    iterative decoding of low density parity check codes based on belief    propagation,” IEEE Trans. Commun., vol. 47, no. 5, pp. 673-680, May    1999.    NPL 6-   J. Chen, A. Dholakia, E. Eleftheriou, M. P. C. Fossorier, and X.-Yu    Hu, “Reduced-complexity decoding of LDPC codes,” IEEE Trans.    Commun., vol. 53, no. 8, pp. 1288-1299, August 2005.    NPL 7-   J. Zhang, and M. P. C. Fossorier, “Shuffled iterative decoding,”    IEEE Trans. Commun., vol. 53, no. 2, pp. 209-213, February 2005.    NPL 8-   S. Lin, D. J. Jr., Costello, “Error control coding: Fundamentals and    applications,” Prentice-Hall.    NPL 9-   Tadashi Wadayama, “Low-Density Parity-Check Code and the decoding    method”, Triceps

SUMMARY OF INVENTION Technical Problem

However, an LDPC-CC, LDPC-CC encoder and LDPC-CC decoder for supportinga plurality of coding rates in a low computational complexity andproviding data of good received quality have not been sufficientlyinvestigated.

For example, Non-Patent Literature 8 discloses using puncturing tosupport a plurality of coding rates. To support a plurality of codingrates using puncturing, first, a basic code (i.e. mother code) isprepared to generate a coding sequence in the mother code and thenselect non-transmission bits (i.e. puncturing bits) from the codingsequence. Further, by changing the number of non-transmission bits, aplurality of coding rates are supported. By this means, it is possibleto support all coding rates by the encoder and decoder (i.e. mother codeencoder and decoder), so that it is possible to provide an advantage ofreducing the computational complexity (i.e. circuit scale).

In contrast, as a method of supporting a plurality of coding rates,there is a method of providing different codes (i.e. distributed codes)every coding rate. Especially, as disclosed in Non-Patent Literature 8,an LDPC code has a flexibility of being able to provide various codelengths and coding rates easily, and therefore it is a general method tosupport a plurality of coding rates by a plurality of codes. In thiscase, although a use of a plurality of codes has a disadvantage ofproviding a large computational complexity (i.e. circuit scale),compared to a case where a plurality of coding rates are supported bypuncturing, there is an advantage of providing data of excellentreceived quality.

In view of the above, there are few documents that argue a method ofgenerating an LDPC code that can maintain the received quality of databy preparing a plurality of codes to support a plurality of codingrates, while reducing the computational complexity of the encoder anddecoder. If a method of providing an LDPC code to realize this isestablished, it is possible to improve the received quality of data andreduce the computational complexity at the same time, which has beendifficult to realize.

Furthermore, an LDPC-CC is a class of a convolutional code, andtherefore requires, for example, termination or tail-biting to securebelief in decoding of information bits. However, studies on an LDPC-CCcapable of minimizing the number of terminations while securingreceiving quality of data, and an encoder and decoder thereof have notbeen carried out sufficiently.

It is therefore an object of the present invention to provide anencoder, decoder and encoding method that can prevent, even whenperforming termination with the encoder and decoder using an LDPC-CC,error correction capability from deteriorating and prevent informationtransmission efficiency from deteriorating.

Solution to Problem

The encoder of the present invention is an encoder that performs LDPC-CCcoding and adopts a configuration including a determining section thatdetermines a sequence length of a termination sequence transmitted bybeing added at a rear end of an information sequence according to aninformation length and coding rate of the information sequence and acomputing section that applies LDPC-CC coding to the informationsequence and a known information sequence necessary to generate thetermination sequence of the determined sequence length, and computes aparity sequence.

The decoder of the present invention is a decoder that decodes anLDPC-CC using belief propagation and adopts a configuration including anacquiring section that acquires a coding rate and a sequence length of atermination sequence transmitted by being added at a rear end of aninformation sequence and a decoding section that performs beliefpropagation decoding on the information sequence based on the codingrate and the termination sequence length.

The encoding method of the present invention determines a sequencelength of a termination sequence transmitted by being added at a rearend of an information sequence according to an information length andcoding rate of the information sequence, applies LDPC-CC coding to theinformation sequence and a known information sequence necessary togenerate the termination sequence of the determined sequence length, andcomputes a parity sequence.

Advantageous Effects of Invention

The encoder, decoder and encoding method of the present invention canprevent, even when performing termination, error correction capabilityfrom deteriorating and prevent information transmission efficiency fromdeteriorating.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 shows an LDPC-CC parity check matrix;

FIG. 2 shows a configuration of an LDPC-CC encoder;

FIG. 3 shows an example of the configuration of an LDPC-CC parity checkmatrix of a time varying period of 4;

FIG. 4A shows parity check polynomials of an LDPC-CC of a time varyingperiod of 3 and the configuration of parity check matrix H of thisLDPC-CC;

FIG. 4B shows the belief propagation relationship of terms relating toX(D) of “check equation #1” to “check equation #3” in FIG. 4A;

FIG. 4C shows the belief propagation relationship of terms relating toX(D) of “check equation #1” to “check equation #6”;

FIG. 5 shows a parity check matrix of a (7, 5) convolutional code;

FIG. 6 shows an example of the configuration of parity check matrix Habout an LDPC-CC of a coding rate of ⅔ and a time varying period of 2;

FIG. 7 shows an example of the configuration of an LDPC-CC parity checkmatrix of a coding rate of ⅔ and a time varying period of m;

FIG. 8 shows an example of the configuration of an LDPC-CC parity checkmatrix of a coding rate of (n−1)/n and a time varying period of m;

FIG. 9 shows an example of the configuration of an LDPC-CC encodingsection;

FIG. 10 is a drawing for explaining a method ofinformation-zero-termination;

FIG. 11 is a block diagram showing the main configuration of an encoderaccording to Embodiment 3 of the present invention;

FIG. 12 is a block diagram showing the main configuration of a firstinformation computing section according to Embodiment 3;

FIG. 13 is a block diagram showing the main configuration of a paritycomputing section according to Embodiment 3;

FIG. 14 is a block diagram showing another main configuration of anencoder according to Embodiment 3;

FIG. 15 is a block diagram showing the main configuration of a decoderaccording to Embodiment 3;

FIG. 16 illustrates operations of a log likelihood ratio setting sectionin a case of a coding rate of ½;

FIG. 17 illustrates operations of a log likelihood ratio setting sectionin a case of a coding rate of ⅔;

FIG. 18 is a diagram showing an example of the configuration of acommunication apparatus equipped with an encoder according to Embodiment3;

FIG. 19 shows an example of a transmission format;

FIG. 20 shows an example of the configuration of a communicationapparatus having a decoder according to Embodiment 3;

FIG. 21 is a diagram showing an example of the relationship between theinformation size and the termination number;

FIG. 22 is a diagram showing another example of the relationship betweenthe information size and the termination number;

FIG. 23 is a diagram showing an example of the relationship between theinformation size and the termination number;

FIG. 24 is a block diagram showing the main configuration of acommunication apparatus having an encoder according to Embodiment 5 ofthe present invention;

FIG. 25 is a diagram illustrating a method of determining a terminationsequence length;

FIG. 26 is a diagram illustrating a method of determining a terminationsequence length;

FIG. 27 shows an example of a transmission format;

FIG. 28 is a block diagram showing the main configuration of acommunication apparatus having a decoder according to Embodiment 5;

FIG. 29 is a diagram showing an example of information flow between thecommunication apparatus having an encoder and the communicationapparatus having a decoder;

FIG. 30 is a diagram showing an example of information flow between thecommunication apparatus having an encoder and the communicationapparatus having a decoder;

FIG. 31 is a diagram showing an example of the table of correspondencebetween the information size and the termination number;

FIG. 32A is a diagram showing BER/BLER characteristics when atermination sequence is added to an information sequence having aninformation size of 512 bits;

FIG. 32B is a diagram showing BER/BLER characteristics when atermination sequence is added to an information sequence having aninformation size of 1024 bits;

FIG. 32C is a diagram showing BER/BLER characteristics when atermination sequence is added to an information sequence having aninformation size of 2048 bits;

FIG. 32D is a diagram showing BER/BLER characteristics when atermination sequence is added to an information sequence having aninformation size of 4096 bits;

FIG. 33 is a diagram showing a table of correspondence between theinformation size and supported coding rates;

FIG. 34 is a block diagram showing the main configuration of acommunication apparatus having an encoder according to Embodiment 6 ofthe present invention;

FIG. 35 is a diagram showing an example of information flow between thecommunication apparatus having an encoder and the communicationapparatus having a decoder;

FIG. 36 is a block diagram showing the main configuration of acommunication apparatus having a decoder according to Embodiment 6;

FIG. 37 is a block diagram showing the main configuration of an encoderaccording to Embodiment 7 of the present invention;

FIG. 38 is a block diagram showing the main configuration of a decoderaccording to Embodiment 7; and

FIG. 39 is a block diagram showing the main configuration of an encoderaccording to Embodiment 8 of the present invention;

DESCRIPTION OF EMBODIMENTS

Now, embodiments of the present invention will be described in detailwith reference to the accompanying drawings

Embodiment 1

First, the present embodiment will describe an LDPC-CC with goodcharacteristics.

(LDPC-CC of Good Characteristics)

An LDPC-CC of a time varying period of g with good characteristics isdescribed below.

First, an LDPC-CC of a time varying period of 4 with goodcharacteristics will be described. A case in which the coding rate is ½is described below as an example.

Consider equations 1-1 to 1-4 as parity check polynomials of an LDPC-CCfor which the time varying period is 4. At this time, X(D) is apolynomial representation of data (information) and P(D) is a paritypolynomial representation. Here, in equations 1-1 to 1-4, parity checkpolynomials have been assumed in which there are four terms in X(D) andP(D) respectively, the reason being that four terms are desirable fromthe standpoint of obtaining good received quality.[1](D ^(a1) +D ^(a2) +D ^(a3) +D ^(a4))X(D)+(D ^(b1) +D ^(b2) +D ^(b3) +D^(b4))P(D)=0  (Equation 1-1)(D ^(A1) +D ^(A2) +D ^(A3) +D ^(A4))X(D)+(D ^(B1) +D ^(B2) +D ^(B3) +D^(B4))P(D)=0  (Equation 1-2)(D ^(α1) +D ^(α2) +D ^(a3) +D ^(α4))X(D)+(D ^(β1) +D ^(β2) +D ^(β3) +D^(β4))P(D)=0  (Equation 1-3)(D ^(E1) +D ^(E2) +D ^(E3) +D ^(E4))X(D)+(D ^(F1) +D ^(F2) +D ^(F3) +D^(F4))P(D)=0  (Equation 1-4)

In equation 1-1, it is assumed that a1, a2, a3 and a4 are integers(where a1≠a2≠a3≠a4, and a1 to a4 are all mutually different). Use of thenotation “X≠Y≠ . . . ≠Z” is assumed to express the fact that X, Y, . . ., Z are all mutually different. Also, it is assumed that b1, b2, b3 andb4 are integers (where b1≠b2≠b3≠b4). A parity check polynomial ofequation 1-1 is called “check equation #1,” and a sub-matrix based onthe parity check polynomial of equation 1-1 is designated firstsub-matrix H1.

In equation 1-2, it is assumed that A1, A2, A3, and A4 are integers(where A1≠A2≠A3≠A4). Also, it is assumed that B1, B2, B3, and B4 areintegers (where B1≠B2≠B3≠B4). A parity check polynomial of equation 1-2is called “check equation #2,” and a sub-matrix based on the paritycheck polynomial of equation 1-2 is designated second sub-matrix H₂.

In equation 1-3, it is assumed that α1, α2, α3, and α4 are integers(where α1≠α2≠α3≠α4). Also, it is assumed that β1, β2, β3, and β4 areintegers (where β1≠β2≠β3≠β4). A parity check polynomial of equation 1-3is called “check equation #3,” and a sub-matrix based on the paritycheck polynomial of equation 1-3 is designated third sub-matrix H₃

In equation 1-4, it is assumed that E1, E2, E3, and E4 are integers(where E1≠E2≠E3≠E4). Also, it is assumed that F1, F2, F3, and F4 areintegers (where F1≠F2≠F3≠F4). A parity check polynomial of equation 1-4is called “check equation #4,” and a sub-matrix based on the paritycheck polynomial of equation 1-4 is designated fourth sub-matrix H₄.

Next, an LDPC-CC of a time varying period of 4 is considered thatgenerates a parity check matrix such as shown in FIG. 3 from firstsub-matrix H₁, second sub-matrix H₂, third sub-matrix H₃, and fourthsub-matrix H₄.

At this time, if k is designated as a remainder after dividing thevalues of combinations of orders of X(D) and P(D), (a1, a2, a3, a4),(b1, b2, b3, b4), (A1, A2, A3, A4), (B1, B2, B3, B4), (α1, α2, α3, α4),(β1, β2, β3, β4), (E1, E2, E3, E4), (F1, F2, F3, F4), in equations 1-1to 1-4 by 4, provision is made for one each of remainders 0, 1, 2, and 3to be included in four-coefficient sets represented as shown above (forexample, (a1, a2, a3, a4)), and to hold true for all the abovefour-coefficient sets.

For example, if orders (a1, a2, a3, a4) of X(D) of “check equation #1”are set as (a1, a2, a3, a4)=(8, 7, 6, 5), remainders k after dividingorders (a1, a2, a3, a4) by 4 are (0, 3, 2, 1), and one each of 0, 1, 2and 3 are included in the four-coefficient set as remainders k.Similarly, if orders (b1, b2, b3, b4) of P(D) of “check equation #1” areset as (b1, b2, b3, b4)=(4, 3, 2, 1), remainders k after dividing orders(b1, b2, b3, b4) by 4 are (0, 3, 2, 1), and one each of 0, 1, 2 and 3are included in the four-coefficient set as remainders k. It is assumedthat the above condition about “remainder” also holds true for thefour-coefficient sets of X(D) and P(D) of the other parity checkequations (“check equation #2,” “check equation #3” and “check equation#4”).

By this means, the column weight of parity check matrix H configuredfrom equations 1-1 to 1-4 becomes 4 for all columns, which enables aregular LDPC code to be formed. Here, a regular LDPC code is an LDPCcode that is defined by a parity check matrix for which each columnweight is equally fixed, and is characterized by the fact that itscharacteristics are stable and an error floor is unlikely to occur. Inparticular, since the characteristics are good when the column weight is4, an LDPC-CC offering good reception performance can be obtained bygenerating an LDPC-CC as described above.

Table 1 shows examples of LDPC-CCs (LDPC-CCs #1 to #3) of a time varyingperiod of 4 and a coding rate of ½ for which the above condition about“remainder” holds true. In table 1, LDPC-CCs of a time varying period of4 are defined by four parity check polynomials: “check polynomial #1,”“check polynomial #2,” “check polynomial #3,” and “check polynomial #4.”

TABLE 1 Code Parity check polynomial LDPC-CC #1 Check polynomial #1:(D⁴⁵⁸ + D⁴³⁵ + D³⁴¹ + 1)X(D) + (D⁵⁹⁸ + D³⁷³ + D⁶⁷ + 1)P(D) = 0 of a timeCheck polynomial #2: (D²⁸⁷ + D²¹³ + D¹³⁰ + 1)X(D) + (D⁵⁴⁵ + D⁵⁴² +D¹⁰³ + 1)P(D) = 0 varying Check polynomial #3: (D⁵⁵⁷ + D⁴⁹⁵ + D³²⁶ +1)X(D) + (D⁵⁶¹ + D⁵⁰² + D³⁵¹ + 1)P(D) = 0 period Check polynomial #4:(D⁴²⁶ + D³²⁹ + D⁹⁹ + 1)X(D) + (D³²¹ + D⁵⁵ + D⁴² + 1)P(D) = 0 of 4 and acoding rate of ½ LDPC-CC #2 Check polynomial #1: (D⁵⁰³ + D⁴⁵⁴ + D⁴⁹ +1)X(D) + (D⁵⁶⁹ + D⁴⁶⁷ + D⁴⁰³ + 1)P(D) = 0 of a time Check polynomial #2:(D⁵¹⁸ + D⁴⁷³ + D²⁰³ + 1)X(D) + (D⁵⁹⁸ + D⁴⁹⁹ + D¹⁴⁵ + 1)P(D) = 0 varyingCheck polynomial #3: (D⁴⁰³ + D³⁹⁷ + D⁶⁸ + 1)X(D) + (D²⁹⁴ + D²⁶⁷ + D⁶⁹ +1)P(D) = 0 period Check polynomial #4: (D⁴⁸³ + D³⁸⁵ + D⁹⁴ + 1)X(D) +(D⁴²⁶ + D⁴¹⁵ + D⁴¹³ + 1)P(D) = 0 of 4 and a coding rate of ½ LDPC-CC #3Check polynomial #1: (D⁴⁵⁴ + D⁴⁴⁷ + D¹⁷ + 1)X(D) + (D⁴⁹⁴ + D²³⁷ + D⁷ +1)P(D) = 0 of a time Check polynomial #2: (D⁵⁸³ + D⁵⁴⁵ + D⁵⁰⁶ + 1)X(D) +(D³²⁵ + D⁷¹ + D⁶⁶ + 1)P(D) = 0 varying Check polynomial #3: (D⁴³⁰ +D⁴²⁵ + D⁴⁰⁷ + 1)X(D) + (D⁵⁸² + D⁴⁷ + D⁴⁵ + 1)P(D) = 0 period Checkpolynomial #4: (D⁴³⁴ + D³⁵³ + D¹²⁷ + 1)X(D) + (D³⁴⁵ + D²⁰⁷ + D³⁸ +1)P(D) = 0 of 4 and a coding rate of ½

In the above description, a case in which the coding rate is ½ has beendescribed as an example, but a regular LDPC code is also formed and goodreceived quality can be obtained when the coding rate is (n−1)/n if theabove condition about “remainder” holds true for four-coefficient setsin information X1(D), X2(D), . . . , Xn−1(D).

In the case of a time varying period of 2, also, it has been confirmedthat a code with good characteristics can be found if the abovecondition about “remainder” is applied. An LDPC-CC of a time varyingperiod of 2 with good characteristics is described below. A case inwhich the coding rate is ½ is described below as an example.

Consider equations 2-1 and 2-2 as parity check polynomials of an LDPC-CCfor which the time varying period is 2. At this time, X(D) is apolynomial representation of data (information) and P(D) is a paritypolynomial representation. Here, in equations 2-1 and 2-2, parity checkpolynomials have been assumed in which there are four terms in X(D) andP(D) respectively, the reason being that four terms are desirable fromthe standpoint of obtaining good received quality.[2](D ^(a1) +D ^(a2) +D ^(a3) +D ^(a4))X(D)+(D ^(b1) +D ^(b2) +D ^(b3) +D^(b4))P(D)=0  (Equation 2-1)(D ^(A1) +D ^(A2) +D ^(A3) +D ^(A4))X(D)+(D ^(B1) +D ^(B2) +D ^(B3) +D^(B4))P(D)=0  (Equation 2-2)

In equation 2-1, it is assumed that a1, a2, a3, and a4 are integers(where a1≠a2≠a3≠a4). Also, it is assumed that b1, b2, b3, and b4 areintegers (where b1≠b2≠b3≠b4). A parity check polynomial of equation 2-1is called “check equation #1,” and a sub-matrix based on the paritycheck polynomial of equation 2-1 is designated first sub-matrix H₁.

In equation 2-2, it is assumed that A1, A2, A3, and A4 are integers(where A1≠A2≠A3≠A4). Also, it is assumed that B1, B2, B3, and B4 areintegers (where B1≠B2≠B3≠B4). A parity check polynomial of equation 2-2is called “check equation #2,” and a sub-matrix based on the paritycheck polynomial of equation 2-2 is designated second sub-matrix H₂.

Next, an LDPC-CC of a time varying period of 2 generated from firstsub-matrix H₁ and second sub-matrix H₂ is considered.

At this time, if k is designated as a remainder after dividing thevalues of combinations of orders of X(D) and P(D), (a1, a2, a3, a4),(b1, b2, b3, b4), (A1, A2, A3, A4), (B1, B2, B3, B4), in equations 2-1and 2-2 by 4, provision is made for one each of remainders 0, 1, 2, and3 to be included in four-coefficient sets represented as shown above(for example, (a1, a2, a3, a4)), and to hold true for all the abovefour-coefficient sets.

For example, if orders (a1, a2, a3, a4) of X(D) of “check equation #1”are set as (a1, a2, a3, a4)=(8, 7, 6, 5), remainders k after dividingorders (a1, a2, a3, a4) by 4 are (0, 3, 2, 1), and one each of 0, 1, 2and 3 are included in the four-coefficient set as remainders k.Similarly, if orders (b1, b2, b3, b4) of P(D) of “check equation #1” areset as (b1, b2, b3, b4)=(4, 3, 2, 1), remainders k after dividing orders(b1, b2, b3, b4) by 4 are (0, 3, 2, 1), and one each of 0, 1, 2 and 3are included in the four-coefficient set as remainders k. It is assumedthat the above condition about “remainder” also holds true for thefour-coefficient sets of X(D) and P(D) of “check equation #2.”

By this means, the column weight of parity check matrix H configuredfrom equations 2-1 and 2-2 becomes 4 for all columns, which enables aregular LDPC code to be formed. Here, a regular LDPC code is an LDPCcode that is defined by a parity check matrix for which each columnweight is equally fixed, and is characterized by the fact that itscharacteristics are stable and an error floor is unlikely to occur. Inparticular, since the characteristics are good when the column weight is8, an LDPC-CC enabling reception performance to be further improved canbe obtained by generating an LDPC-CC as described above.

Table 2 shows examples of LDPC-CCs (LDPC-CCs #1 and #2) of a timevarying period of 2 and a coding rate of ½ for which the above conditionabout “remainder” holds true. In table 2, LDPC-CCs of a time varyingperiod of 2 are defined by two parity check polynomials: “checkpolynomial #1” and “check polynomial #2.”

TABLE 2 Code Parity check polynomial LDPC-CC #1 Check polynomial #1:(D⁵⁵¹ + D⁴⁶⁵ + D⁹⁸ + 1)X(D) + of a time (D⁴⁰⁷ + D³⁸⁶ + D³⁷³ + 1)P(D) = 0varying Check polynomial #2: (D⁴⁴³ + D⁴³³ + D⁵⁴ + 1)X(D) + period(D⁵⁵⁹ + D⁵⁵⁷ + D⁵⁴⁶ + 1)P(D) = 0 of 2 and a coding rate of ½ LDPC-CC #2Check polynomial #1: (D²⁶⁵ + D¹⁹⁰ + D⁹⁹ + 1)X(D) + of a time (D²⁹⁵ +D²⁴⁶ + D⁶⁹ + 1)P(D) = 0 varying Check polynomial #2: (D²⁷⁵ + D²²⁶ +D²¹³ + 1)X(D) + period 0 (D²⁹⁸ + D¹⁴⁷ + D⁴⁵ + 1)P(D) = 0 of 2 and acoding rate of ½

In the above description (LDPC-CCs of a time varying period of 2), acase in which the coding rate is ½ has been described as an example, buta regular LDPC code is also formed and good received quality can beobtained when the coding rate is (n−1)/n if the above condition about“remainder” holds true for four-coefficient sets in information X1(D),X2(D), . . . , Xn−1(D).

In the case of a time varying period of 3, also, it has been confirmedthat a code with good characteristics can be found if the followingcondition about “remainder” is applied. An LDPC-CC of a time varyingperiod of 3 with good characteristics is described below. A case inwhich the coding rate is ½ is described below as an example.

Consider equations 3-1 to 3-3 as parity check polynomials of an LDPC-CCfor which the time varying period is 3. At this time, X(D) is apolynomial representation of data (information) and P(D) is a paritypolynomial representation. Here, in equations 3-1 to 3-3, parity checkpolynomials are assumed such that there are three terms in X(D) and P(D)respectively.[3](D ^(a1) +D ^(a2) +D ^(a3))X(D)+(D ^(b1) +D ^(b2) +D^(b3))P(D)=0  (Equation 3-1)(D ^(A1) +D ^(A2) +D ^(A3))X(D)+(D ^(B1) +D ^(B2) +D^(B3))P(D)=0  (Equation 3-2)(D ^(α1) +D ^(α2) +D ^(a3))X(D)+(D ^(β1) +D ^(β2) +D^(b3))P(D)=0  (Equation 3-3)

In equation 3-1, it is assumed that a1, a2, and a3 are integers (wherea1≠a2≠a3). Also, it is assumed that b1, b2 and b3 are integers (whereb1≠b2≠b3). A parity check polynomial of equation 3-1 is called “checkequation #1,” and a sub-matrix based on the parity check polynomial ofequation 3-1 is designated first sub-matrix H₁.

In equation 3-2, it is assumed that A1, A2 and A3 are integers (whereA1≠A2≠A3). Also, it is assumed that B1, B2 and B3 are integers (whereB1≠B2≠B3). A parity check polynomial of equation 3-2 is called “checkequation #2,” and a sub-matrix based on the parity check polynomial ofequation 3-2 is designated second sub-matrix H₂.

In equation 3-3, it is assumed that α1, α2 and α3 are integers (whereα1≠α2≠α3). Also, it is assumed that β1, β2 and β3 are integers (whereβ1≠β2≠β3). A parity check polynomial of equation 3-3 is called “checkequation #3,” and a sub-matrix based on the parity check polynomial ofequation 3-3 is designated third sub-matrix H₃.

Next, an LDPC-CC of a time varying period of 3 generated from firstsub-matrix H₁, second sub-matrix H₂ and third sub-matrix H₃ isconsidered.

At this time, if k is designated as a remainder after dividing thevalues of combinations of orders of X(D) and P(D), (a1, a2, a3), (b1,b2, b3), (A1, A2, A3), (B1, B2, B3), (α1, α2, α3), (β1, β2, β3), inequations 3-1 to 3-3 by 3, provision is made for one each of remainders0, 1, and 2 to be included in three-coefficient sets represented asshown above (for example, (a1, a2, a3)), and to hold true for all theabove three-coefficient sets.

For example, if orders (a1, a2, a3) of X(D) of “check equation #1” areset as (a1, a2, a3)=(6, 5, 4), remainders k after dividing orders (a1,a2, a3) by 3 are (0, 2, 1), and one each of 0, 1, 2 are included in thethree-coefficient set as remainders k. Similarly, if orders (b1, b2, b3)of P(D) of “check equation #1” are set as (b1, b2, b3)=(3, 2, 1),remainders k after dividing orders (b1, b2, b3) by 3 are (0, 2, 1), andone each of 0, 1, 2 are included in the three-coefficient set asremainders k. It is assumed that the above condition about “remainder”also holds true for the three-coefficient sets of X(D) and P(D) of“check equation #2” and “check equation #3.”

By generating an LDPC-CC as above, it is possible to generate a regularLDPC-CC code in which the row weight is equal in all rows and the columnweight is equal in all columns, without some exceptions. Here,“exceptions” refer to part in the beginning of a parity check matrix andpart in the end of the parity check matrix, where the row weights andcolumns weights are not the same as row weights and column weights ofthe other part. Furthermore, when BP decoding is performed, belief in“check equation #2” and belief in “check equation #3” are propagatedaccurately to “check equation #1,” belief in “check equation #1” andbelief in “check equation #3” are propagated accurately to “checkequation #2,” and belief in “check equation #1” and belief in “checkequation #2” are propagated accurately to “check equation #3.”Consequently, an LDPC-CC with better received quality can be obtained.This is because, when considered in column units, positions at which “1”is present are arranged so as to propagate belief accurately, asdescribed above.

The above belief propagation will be described below using accompanyingdrawings. FIG. 4A shows parity check polynomials of an LDPC-CC of a timevarying period of 3 and the configuration of parity check matrix H ofthis LDPC-CC.

“Check equation #1” illustrates a case in which (a1, a2, a3)=(2, 1, 0)and (b1, b2, b3)=(2, 1, 0) in a parity check polynomial of equation 3-1,and remainders after dividing the coefficients by 3 are as follows:(a1%3, a2%3, a3%3)=(2, 1, 0), (b1%3, b2%3, b3%3)=(2, 1, 0), where “Z %3”represents a remainder after dividing Z by 3 (the same applieshereinafter).

“Check equation #2” illustrates a case in which (A1, A2, A3)=(5, 1, 0)and (B1, B2, B3)=(5, 1, 0) in a parity check polynomial of equation 3-2,and remainders after dividing the coefficients by 3 are as follows:(A1%3, A2%3, A3%3)=(2, 1, 0), (B1%3, B2%3, B3%3)=(2, 1, 0).

“Check equation #3” illustrates a case in which (α1, α2, α3)=(4, 2, 0)and (β1, β2, β3)=(4, 2, 0) in a parity check polynomial of equation 3-3,and remainders after dividing the coefficients by 3 are as follows:(α1%3, α2%3, α3%3)=(1, 2, 0), (β1%3, β2%3, β3%3)=(1, 2, 0).

Therefore, the example of LDPC-CC of a time varying period of 3 shown inFIG. 4A satisfies the above condition about “remainder”, that is, acondition that (a1%3, a2%3, a3%3), (b1%3, b2%3, b3%3), (A1%3, A2%3,A3%3), (B1%3, B2%3, B3%3), (α1%3, α2%3, α3%3) and (β1%3, β2%3, β3%3) areany of the following: (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0,1), (2, 1, 0).

Returning to FIG. 4A again, belief propagation will now be explained. Bycolumn computation of column 6506 in BP decoding, for “1” of area 6501of “check equation #1,” belief is propagated from “1” of area 6504 of“check equation #2” and from “1” of area 6505 of “check equation #3.” Asdescribed above, “1” of area 6501 of “check equation #1” is acoefficient for which a remainder after division by 3 is 0 (a3%3=0(a3=0) or b3%3=0 (b3=0)). Also, “1” of area 6504 of “check equation #2”is a coefficient for which a remainder after division by 3 is 1 (A2%3=1(A2=1) or B2%3=1 (B2=1)). Furthermore, “1” of area 6505 of “checkequation #3” is a coefficient for which a remainder after division by 3is 2 (α2%3=2 (α2=2) or β2%3=2 (β2=2)).

Thus, for “1” of area 6501 for which a remainder is 0 in thecoefficients of “check equation #1,” in column computation of column6506 in BP decoding, belief is propagated from “1” of area 6504 forwhich a remainder is 1 in the coefficients of “check equation #2” andfrom “1” of area 6505 for which a remainder is 2 in the coefficients of“check equation #3.”

Similarly, for “1” of area 6502 for which a remainder is 1 in thecoefficients of “check equation #1,” in column computation of column6509 in BP decoding, belief is propagated from “1” of area 6507 forwhich a remainder is 2 in the coefficients of “check equation #2” andfrom “1” of area 6508 for which a remainder is 0 in the coefficients of“check equation #3.”

Similarly, for “1” of area 6503 for which a remainder is 2 in thecoefficients of “check equation #1,” in column computation of column6512 in BP decoding, belief is propagated from “1” of area 6510 forwhich a remainder is 0 in the coefficients of “check equation #2” andfrom “1” of area 6511 for which a remainder is 1 in the coefficients of“check equation #3.”

A supplementary explanation of belief propagation will now be givenusing FIG. 4B. FIG. 4B shows the belief propagation relationship ofterms relating to X(D) of “check equation #1” to “check equation #3” inFIG. 4A. “Check equation #1” to “check equation #3” in FIG. 4Aillustrate cases in which (a1, a2, a3)=(2, 1, 0), (A1, A2, A3)=(5, 1,0), and (α1, α2, α3)=(4, 2, 0), in terms relating to X(D) of equations3-1 to 3-3.

In FIG. 4B, terms (a3, A3, α3) inside squares indicate coefficients forwhich a remainder after division by 3 is 0, terms (a2, A2, α2) insidecircles indicate coefficients for which a remainder after division by 3is 1, and terms (a1, A1, α1) inside diamond-shaped boxes indicatecoefficients for which a remainder after division by 3 is 2.

As can be seen from FIG. 4B, for a1 of “check equation #1,” belief ispropagated from A3 of “check equation #2” and from α1 of “check equation#3” for which remainders after division by 3 differ; for a2 of “checkequation #1,” belief is propagated from A1 of “check equation #2” andfrom α3 of “check equation #3” for which remainders after division by 3differ; and, for a3 of “check equation #1,” belief is propagated from A2of “check equation #2” and from α2 of “check equation #3” for whichremainders after division by 3 differ. While FIG. 4B shows the beliefpropagation relationship of terms relating to X(D) of “check equation#1” to “check equation #3,” the same applies to terms relating to P(D).

Thus, for “check equation #1,” belief is propagated from coefficientsfor which remainders after division by 3 are 0, 1, and 2 amongcoefficients of “check equation #2.” That is to say, for “check equation#1,” belief is propagated from coefficients for which remainders afterdivision by 3 are all different among coefficients of “check equation#2.” Therefore, beliefs with low correlation are all propagated to“check equation #1.”

Similarly, for “check equation #2,” belief is propagated fromcoefficients for which remainders after division by 3 are 0, 1, and 2among coefficients of “check equation #1.” That is to say, for “checkequation #2,” belief is propagated from coefficients for whichremainders after division by 3 are all different among coefficients of“check equation #1.” Also, for “check equation #2,” belief is propagatedfrom coefficients for which remainders after division by 3 are 0, 1, and2 among coefficients of “check equation #3.” That is to say, for “checkequation #2,” belief is propagated from coefficients for whichremainders after division by 3 are all different among coefficients of“check equation #3.”

Similarly, for “check equation #3,” belief is propagated fromcoefficients for which remainders after division by 3 are 0, 1, and 2among coefficients of “check equation #1.” That is to say, for “checkequation #3,” belief is propagated from coefficients for whichremainders after division by 3 are all different among coefficients of“check equation #1.” Also, for “check equation #3,” belief is propagatedfrom coefficients for which remainders after division by 3 are 0, 1, and2 among coefficients of “check equation #2.” That is to say, for “checkequation #3,” belief is propagated from coefficients for whichremainders after division by 3 are all different among coefficients of“check equation #2.”

By providing for the orders of parity check polynomials of equations 3-1to 3-3 to satisfy the above condition about “remainder” in this way,belief is necessarily propagated in all column computations, so that itis possible to perform belief propagation efficiently in all checkequations and further increase error correction capability.

A case in which the coding rate is ½ has been described above for anLDPC-CC of a time varying period of 3, but the coding rate is notlimited to ½. A regular LDPC code is also formed and good receivedquality can be obtained when the coding rate is (n−1)/n (where n is aninteger equal to or greater than 2) if the above condition about“remainder” holds true for three-coefficient sets in information X1(D),X2(D), . . . , Xn−1(D).

A case in which the coding rate is (n−1)/n (where n is an integer equalto or greater than 2) is described below.

Consider equations 4-1 to 4-3 as parity check polynomials of an LDPC-CCfor which the time varying period is 3. At this time, X₁(D), X₂(D), . .. , X_(n-1)(D) are polynomial representations of data (information) X₁,X₂, . . . , X_(n-1), and P(D) is a polynomial representation of parity.Here, in equations 4-1 to 4-3, parity check polynomials are assumed suchthat there are three terms in X₁(D), X₂(D), . . . , X_(n-1)(D), and P(D)respectively.[4](D ^(a1,1) +D ^(a1,2) +D ^(a1,3))X ₁(D)+(D ^(a2,1) +D ^(a2,2) +D^(a2,3))X ₂(D)+ . . . +(D ^(an-1,1) +D ^(an-1,2) +D ^(an-1,3))X_(n-1)(D)+(D ^(b1) +D ^(b2) +D ^(b3))P(D)=0  (Equation 4-1)(D ^(A1,1) +D ^(A1,2) +D ^(A1,3))X ₁(D)+(D ^(A2,1) +D ^(A2,2) +D^(A2,3))X ₂(D)+ . . . +(D ^(An-1,1) +D ^(An-1,2) +D ^(An-1,3))X_(n-1)(D)+(D ^(B1) +D ^(B2) +D ^(B3))P(D)=0  (Equation 4-2).(D ^(α1,1) +D ^(α1,2) +D ^(α1,3))X ₁(D)+(D ^(α2,1) +D ^(α2,2) +D^(α2,3))X ₂(D)+ . . . +(D ^(αn-1,1) +D ^(αn-1,2) +D ^(αn-1,3))X_(n-1)(D)+(D ^(β1) +D ^(β2) +D ^(β3))P(D)=0  (Equation 4-3)

In equation 4-1, it is assumed that a_(i,1), a_(i,2), and a_(i,3) (wherei=1, 2, . . . , n−1) are integers (where a_(i,1)≠a_(i,2)≠a_(i,3)). Also,it is assumed that b1, b2 and b3 are integers (where b1≠b2≠b3). A paritycheck polynomial of equation 4-1 is called “check equation #1,” and asub-matrix based on the parity check polynomial of equation 4-1 isdesignated first sub-matrix H₁.

In equation 4-2, it is assumed that A_(i,1), A_(i,2), and A_(i,3) (wherei=1, 2, . . . , n−1) are integers (where A_(i,1)≠A_(i,2)≠A_(i,3)). Also,it is assumed that B1, B2 and B3 are integers (where B1≠B2≠B3). A paritycheck polynomial of equation 4-2 is called “check equation #2,” and asub-matrix based on the parity check polynomial of equation 4-2 isdesignated second sub-matrix H₂.

In equation 4-3, it is assumed that α_(i,1), α_(i,2), and α_(i,3) (wherei=1, 2, . . . , n−1) are integers (where α_(i,1)≠α_(i,2)≠α_(i,3)). Also,it is assumed that β1, β2 and β3 are integers (where β1≠β2≠β3). A paritycheck polynomial of equation 4-3 is called “check equation #3,” and asub-matrix based on the parity check polynomial of equation 4-3 isdesignated third sub-matrix H₃.

Next, an LDPC-CC of a time varying period of 3 generated from firstsub-matrix H₁, second sub-matrix H₂ and third sub-matrix H₃ isconsidered.

At this time, if k is designated as a remainder after dividing thevalues of combinations of orders of X₁(D), X₂(D), . . . , X_(n-1)(D),and P(D),

(a_(1,1), a_(1,2), a_(1,3)),

a_(2,1), a_(2,2), a_(2,3)), . . . ,

(a_(n-1), a_(n-1,2), a_(n-1,3)),

(b1, b2, b3),

(A_(1,1), A_(1,2), A_(1,3)),

A_(2,1), A_(2,2), A_(2,3)), . . . ,

(A_(n-1,1), A_(n-1,2), A_(n-1,3)),

(B1, B2, B3),

(α_(1,1), α_(1,2), α_(1,3)),

(α_(2,1), α_(2,2), α_(2,3)), . . . ,

(α_(n-1,1), α_(n-1,2), α_(n-1,3)), (β1, β2, β3),

in equations 4-1 to 4-3 by 3, provision is made for one each ofremainders 0, 1, and 2 to be included in three-coefficient setsrepresented as shown above (for example, (a_(1,1), a_(1,2), a_(1,3))),and to hold true for all the above three-coefficient sets.

That is to say, provision is made for

(a_(1,1)%3, a_(1,2)%3, a_(1,3)%3),

(a_(2,1)%3, a_(2,2)%3, a_(2,3)%3), . . . ,

(a_(n-1,1)%3, a_(n-1,2)%3, a_(n-1,3)%3),

(b1%3, b2%3, b3%3),

(A_(1,1)%3, A_(1,2)%3, A_(1,3)%3),

(A2,1%3, A2,2%3, A_(2,3)%3), . . . ,

(A_(n-1,1)%3, A_(n-1,2)%3, A_(n-1,3)%3),

(B1%3, B2%3, B3%3),

(α_(1,1)%3, α_(1,2)%3, α_(1,3)%3),

(α_(2,1)%3, α_(2,2)%3, α_(2,3)%3), . . . ,

(α_(n-1,1)%3, α_(n-1,2)%3, α_(n-1,3)%3), and

(β1%3, β2%3, β3%3) to be any of the following: (0, 1, 2), (0, 2, 1), (1,0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0).

Generating an LDPC-CC in this way enables a regular LDPC-CC code to begenerated. Furthermore, when BP decoding is performed, belief in “checkequation #2” and belief in “check equation #3” are propagated accuratelyto “check equation #1,” belief in “check equation #1” and belief in“check equation #3” are propagated accurately to “check equation #2,”and belief in “check equation #1” and belief in “check equation #2” arepropagated accurately to “check equation #3.” Consequently, an LDPC-CCwith better received quality can be obtained in the same way as in thecase of a coding rate of ½.

Table 3 shows examples of LDPC-CCs (LDPC-CCs #1, #2, #3, #4, #5 and #6)of a time varying period of 3 and a coding rate of ½ for which the above“remainder” related condition holds true. In table 3, LDPC-CCs of a timevarying period of 3 are defined by three parity check polynomials:“check (polynomial) equation #1,” “check (polynomial) equation #2” and“check (polynomial) equation #3.”

TABLE 3 Code Parity check polynomial LDPC-CC #1 Check polynomial #1:(D⁴²⁸ + D³²⁵ + 1)X(D) + (D⁵³⁸ + D³³² + 1)P(D) = 0 of a time Checkpolynomial #2: (D⁵³⁸ + D³⁸⁰ + 1)X(D) + (D⁴⁴⁹ + D¹ + 1)P(D) = 0 varyingperiod Check polynomial #3: (D⁵⁸³ + D¹⁷⁰ + 1)X(D) + (D³⁰⁴ + D²⁴² +1)P(D) = 0 of 3 and a coding rate of ½ LDPC-CC #2 Check polynomial #1:(D⁵⁶² + D⁷¹ + 1)X(D) + (D³²⁵ + D¹³⁵ + 1)P(D) = 0 of a time Checkpolynomial #2: (D²¹⁵ + D¹⁰⁶ + 1)X(D) + (D⁵⁶⁶ + D¹⁴² + 1)P(D) = 0 varyingperiod Check polynomial #3: (D⁵⁹⁰ + D⁵⁵⁹ + 1)X(D) + (D¹²⁷ + D¹¹⁰ +1)P(D) = 0 of 3 and a coding rate of ½ LDPC-CC #3 Check polynomial #1:(D¹¹² + D⁵³ + 1)X(D) + (D¹¹⁰ + D⁸⁸ + 1)P(D) = 0 of a time Checkpolynomial #2: (D¹⁰³ + D⁴⁷ + 1)X(D) + (D⁸⁵ + D⁸³ + 1)P(D) = 0 varyingperiod Check polynomial #3: (D¹⁴⁸ + D⁸⁹ + 1)X(D) + (D¹⁴⁶ + D⁴⁹ + 1)P(D)= 0 of 3 and a coding rate of ½ LDPC-CC #4 Check polynomial #1: (D³⁵⁰ +D³²² + 1)X(D) + (D⁴⁴⁸ + D³³⁸ + 1)P(D) = 0 of a time Check polynomial #2:(D⁵²⁹ + D³² + 1)X(D) + (D²³⁸ + D¹⁸⁸ + 1)P(D) = 0 varying period Checkpolynomial #3: (D⁵⁹² + D⁵⁷² + 1)X(D) + (D⁵⁷⁸ + D⁵⁶⁸ + 1)P(D) = 0 of 3and a coding rate of ½ LDPC-CC #5 Check polynomial #1: (D⁴¹⁰ + D⁸² +1)X(D) + (D⁸³⁵ + D⁴⁷ + 1)P(D) = 0 of a time Check polynomial #2: (D⁸⁷⁵ +D⁷⁹⁶ + 1)X(D) + (D⁹⁶² + D⁸⁷¹ + 1)P(D) = 0 varying period Checkpolynomial #3: (D⁶⁰⁵ + D⁵⁴⁷ + 1)X(D) + (D⁹⁵⁰ + D⁴³⁹ + 1)P(D) = 0 of 3and a coding rate of ½ LDPC-CC #6 Check polynomial #1: (D³⁷³ + D⁵⁶ +1)X(D) + (D⁴⁰⁶ + D²¹⁸ + 1)P(D) = 0 of a time Check polynomial #2:(D⁴⁵⁷ + D¹⁹⁷ + 1)X(D) + (D⁴⁹¹ + D²² + 1)P(D) = 0 varying period Checkpolynomial #3: (D⁴⁸⁵ + D⁷⁰ + 1)X(D) + (D²³⁶ + D¹⁸¹ + 1)P(D) = 0 of 3 anda coding rate of ½

It has been confirmed that, as in the case of a time varying period of3, a code with good characteristics can be found if the condition about“remainder” below is applied to an LDPC-CC for which the time varyingperiod is a multiple of 3 (for example, 6, 9, 12, . . . ). An LDPC-CC ofa multiple of a time varying period of 3 with good characteristics isdescribed below. The case of an LDPC-CC of a coding rate of ½ and a timevarying period of 6 is described below as an example.

Consider equations 5-1 to 5-6 as parity check polynomials of an LDPC-CCfor which the time varying period is 6.[5](D ^(a1,1) +D ^(a1,2) +D ^(a1,3))X(D)+(D ^(b1,1) +D ^(b1,2) +D^(b1,3))P(D)=0  (Equation 5-1)(D ^(a2,1) +D ^(a2,2) +D ^(a2,3))X(D)+(D ^(b2,1) +D ^(b2,2) +D^(b2,3))P(D)=0  (Equation 5-2)(D ^(a3,1) +D ^(a3,2) +D ^(a3,3))X(D)+(D ^(b3,1) +D ^(b3,2) +D^(b3,3))P(D)=0  (Equation 5-3)(D ^(a4,1) +D ^(a4,2) +D ^(a4,3))X(D)+(D ^(b4,1) +D ^(b4,2) +D^(b4,3))P(D)=0  (Equation 5-4)(D ^(a5,1) +D ^(a5,2) +D ^(a5,3))X(D)+(D ^(b5,1) +D ^(b5,2) +D^(b5,3))P(D)=0  (Equation 5-5)(D ^(a6,1) +D ^(a6,2) +D ^(a6,3))X(D)+(D ^(b6,1) +D ^(b6,2) +D^(b6,3))P(D)=0  (Equation 5-6)

At this time, X(D) is a polynomial representation of data (information)and P(D) is a parity polynomial representation. With an LDPC-CC of atime varying period of 6, if i %6=k (where k=0, 1, 2, 3, 4, 5) isassumed for parity Pi and information Xi at time i, a parity checkpolynomial of equation 5−(k+1) holds true. For example, if i=1, i %6=1(k=1), and therefore equation 6 holds true.[6]D ^(a2,1) +D ^(a2,2) +D ^(a2,3))X ₁+(D ^(b2,1) +D ^(b2,2) +D ^(b2,3))P₁=0  (Equation 6)

Here, in equations 5-1 to 5-6, parity check polynomials are assumed suchthat there are three terms in X(D) and P(D) respectively.

In equation 5-1, it is assumed that a1,1, a1,2, a1,3 are integers (wherea1,1≠a1,2≠a1,3). Also, it is assumed that b1,1, b1,2, and b1,3 areintegers (where b1,1≠b1,2≠b1,3). A parity check polynomial of equation5-1 is called “check equation #1,” and a sub-matrix based on the paritycheck polynomial of equation 5-1 is designated first sub-matrix H₁.

In equation 5-2, it is assumed that a2,1, a2,2, and a2,3 are integers(where a2,1≠a2,2≠a2,3). Also, it is assumed that b2,1, b2,2, b2,3 areintegers (where b2,1≠b2,2≠b2,3). A parity check polynomial of equation5-2 is called “check equation #2,” and a sub-matrix based on the paritycheck polynomial of equation 5-2 is designated second sub-matrix H₂.

In equation 5-3, it is assumed that a3,1, a3,2, and a3,3 are integers(where a3,1≠a3,2≠a3,3). Also, it is assumed that b3,1, b3,2, and b3,3are integers (where b3,1≠3,2≠b3,3). A parity check polynomial ofequation 5-3 is called “check equation #3,” and a sub-matrix based onthe parity check polynomial of equation 5-3 is designated thirdsub-matrix H₃.

In equation 5-4, it is assumed that a4,1, a4,2, and a4,3 are integers(where a4,1≠a4,2≠a4,3). Also, it is assumed that b4,1, b4,2, and b4,3are integers (where b4,1≠b4,2≠b4,3). A parity check polynomial ofequation 5-4 is called “check equation #4,” and a sub-matrix based onthe parity check polynomial of equation 5-4 is designated fourthsub-matrix H₄.

In equation 5-5, it is assumed that a5,1, a5,2, and a5,3 are integers(where a5, 1≠a5,2≠a5,3). Also, it is assumed that b5,1, b5,2, and b5,3are integers (where b5,1≠b5,2≠b5,3). A parity check polynomial ofequation 5-5 is called “check equation #5,” and a sub-matrix based onthe parity check polynomial of equation 5-5 is designated fifthsub-matrix H₅.

In equation 5-6, it is assumed that a6,1, a6,2, and a6,3 are integers(where a6,1≠a6,2≠a6,3). Also, it is assumed that b6,1, b6,2, and b6,3are integers (where b6,1≠b6,2≠b6,3). A parity check polynomial ofequation 5-6 is called “check equation #6,” and a sub-matrix based onthe parity check polynomial of equation 5-6 is designated sixthsub-matrix H₆.

Next, an LDPC-CC of a time varying period of 6 is considered that isgenerated from first sub-matrix H₁, second sub-matrix H₂, thirdsub-matrix H₃, fourth sub-matrix H₄, fifth sub-matrix H₅ and sixthsub-matrix H₆.

At this time, if k is designated as a remainder after dividing thevalues of combinations of orders of X(D) and P(D),

(a1,1, a1,2, a1,3),

(b1,1, b1,2, b1,3),

(a2,1, a2,2, a2,3),

(b2,1, b2,2, b2,3),

(a3,1, a3,2, a3,3),

(b3,1, b3,2, b3,3),

(a4,1, a4,2, a4,3),

(b4,1, b4,2, b4,3),

(a5,1, a5,2, a5,3),

(b5,1, b5,2, b5,3),

(a6,1, a6,2, a6,3),

(b6,1, b6,2, b6,3), in equations 5-1 to 5-6 by 3, provision is made forone each of remainders 0, 1, and 2 to be included in three-coefficientsets represented as shown above (for example, (a1,1, a1,2, a1,3)), andto hold true for all the above three-coefficient sets. That is to say,provision is made for

(a1,1%3, a1,2%3, a1,3%3),

(b1,1%3, b1,2%3, b1,3%3),

(a2,1%3, a2,2%3, a2,3%3),

(b2,1%3, b2,2%3, b2,3%3),

(a3,1%3, a3,2%3, a3,3%3),

(b3,1%3, b3,2%3, b3,3%3),

(a4,1%3, a4,2%3, a4,3%3),

(b4,1%3, b4,2%3, b4,3%3),

(a5,1%3, a5,2%3, a5,3%3),

(b5,1%3, b5,2%3, b5,3%3),

(a6,1%3, a6,2%3, a6,3%3) and

(b6,1%3, b6,2%3, b6,3%3) to be any of the following: (0, 1, 2), (0, 2,1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0).

By generating an LDPC-CC in this way, if an edge is present when aTanner graph is drawn for “check equation #1,” belief in “check equation#2 or check equation #5” and belief in “check equation #3 or checkequation #6” are propagated accurately.

Also, if an edge is present when a Tanner graph is drawn for “checkequation #2,” belief in “check equation #1 or check equation #4” andbelief in “check equation #3 or check equation #6” are propagatedaccurately.

If an edge is present when a Tanner graph is drawn for “check equation#3,” belief in “check equation #1 or check equation #4” and belief in“check equation #2 or check equation #5” are propagated accurately. Ifan edge is present when a Tanner graph is drawn for “check equation #4,”belief in “check equation #2 or check equation #5” and belief in “checkequation #3 or check equation #6” are propagated accurately.

If an edge is present when a Tanner graph is drawn for “check equation#5,” belief in “check equation #1 or check equation #4” and belief in“check equation #3 or check equation #6” are propagated accurately. Ifan edge is present when a Tanner graph is drawn for “check equation #6,”belief in “check equation #1 or check equation #4” and belief in “checkequation #2 or check equation #5” are propagated accurately.

Consequently, an LDPC-CC of a time varying period of 6 can maintainbetter error correction capability in the same way as when the timevarying period is 3.

In this regard, belief propagation will be described using FIG. 4C. FIG.4C shows the belief propagation relationship of terms relating to X(D)of “check equation #1” to “check equation #6.” In FIG. 4C, a squareindicates a coefficient for which a remainder after division by 3 inax,y (where x=1, 2, 3, 4, 5, 6, and y=1, 2, 3) is 0.

A circle indicates a coefficient for which a remainder after division by3 in ax,y (where x=1, 2, 3, 4, 5, 6, and y=1, 2, 3) is 1. Adiamond-shaped box indicates a coefficient for which a remainder afterdivision by 3 in ax,y (where x=1, 2, 3, 4, 5, 6, and y=1, 2, 3) is 2.

As can be seen from FIG. 4C, if an edge is present when a Tanner graphis drawn, for a1,1 of “check equation #1,” belief is propagated from“check equation #2 or #5” and “check equation #3 or #6” for whichremainders after division by 3 differ. Similarly, if an edge is presentwhen a Tanner graph is drawn, for a1.2 of “check equation #1,” belief ispropagated from “check equation #2 or #5” and “check equation #3 or #6”for which remainders after division by 3 differ.

Similarly, if an edge is present when a Tanner graph is drawn, for a1,3of “check equation #1,” belief is propagated from “check equation #2 or#5” and “check equation #3 or #6” for which remainders after division by3 differ. While FIG. 4C shows the belief propagation relationship ofterms relating to X(D) of “check equation #1” to “check equation #6,”the same applies to terms relating to P(D).

Thus, belief is propagated to each node in a Tanner graph of “checkequation #1” from coefficient nodes of other than “check equation #1.”Therefore, beliefs with low correlation are all propagated to “checkequation #1,” enabling an improvement in error correction capability tobe expected.

In FIG. 4C, “check equation #1” has been focused upon, but a Tannergraph can be drawn in a similar way for “check equation #2” to “checkequation #6,” and belief is propagated to each node in a Tanner graph of“check equation #K” from coefficient nodes of other than “check equation#K.” Therefore, beliefs with low correlation are all propagated to“check equation #K” (where K=2, 3, 4, 5, 6), enabling an improvement inerror correction capability to be expected.

By providing for the orders of parity check polynomials of equations 5-1to 5-6 to satisfy the above condition about “remainder” in this way,belief can be propagated efficiently in all check equations, and thepossibility of being able to further improve error correction capabilityis increased.

A case in which the coding rate is ½ has been described above for anLDPC-CC of a time varying period of 6, but the coding rate is notlimited to ½. The possibility of obtaining good received quality can beincreased when the coding rate is (n−1)/n (where n is an integer equalto or greater than 2) if the above condition about “remainder” holdstrue for three-coefficient sets in information X₁(D), X₂(D), . . . ,X_(n-1)(D).

A case in which the coding rate is (n−1)/n (where n is an integer equalto or greater than 2) is described below.

Consider equations 7-1 to 7-6 as parity check polynomials of an LDPC-CCfor which the time varying period is 6.[7](D ^(a#1,1,1) +D ^(a#1,1,2) +D ^(a#1,1,3))X ₁(D)+(D ^(a#1,2,1) +D^(a#1,2,2) +D ^(a#1,2,3))X ₂(D)+ . . . +(D ^(a#1,n-1,1) +D ^(a#1,n-1,2)+D ^(a#1,n-1,3))X _(n-1)(D)+(D ^(b#1,1) +D ^(b#1,2) +D^(b#1,3))P(D)=0  (Equation 7-1)(D ^(a#2,1,1) +D ^(a#2,1,2) +D ^(a#2,1,3))X ₁(D)+(D ^(a#2,2,1) +D^(a#2,2,2) +D ^(a#2,2,3))X ₂(D)+ . . . +(D ^(a#2,n-1,1) +D ^(a#2,n-1,2)+D ^(a#2,n-1,3))X _(n-1)(D)+(D ^(b#2,1) +D ^(b#2,2) +D^(b#2,3))P(D)=0  (Equation 7-2)(D ^(a#3,1,1) +D ^(a#3,1,2) +D ^(a#3,1,3))X ₁(D)+(D ^(a#3,2,1) +D^(a#3,2,2) +D ^(a#3,2,3))X ₂(D)+ . . . +(D ^(a#3,n-1,1) +D ^(a#3,n-1,2)+D ^(a#3,n-1,3))X _(n-1)(D)+(D ^(b#3,1) +D ^(b#3,2) +D^(b#3,3))P(D)=0  (Equation 7-3)(D ^(a#4,1,1) +D ^(a#4,1,2) +D ^(a#4,1,3))X ₁(D)+(D ^(a#4,2,1) +D^(a#4,2,2) +D ^(a#4,2,3))X ₂(D)+ . . . +(D ^(a#4,n-1,1) +D ^(a#4,n-1,2)+D ^(a#4,n-1,3))X _(n-1)(D)+(D ^(b#4,1) +D ^(b#4,2) +D^(b#4,3))P(D)=0  (Equation 7-4)(D ^(a#5,1,1) +D ^(a#5,1,2) +D ^(a#5,1,3))X ₁(D)+(D ^(a#5,2,1) +D^(a#5,2,2) +D ^(a#5,2,3))X ₂(D)+ . . . +(D ^(a#5,n-1,1) +D ^(a#5,n-1,2)+D ^(a#5,n-1,3))X _(n-1)(D)+(D ^(b#5,1) +D ^(b#5,2) +D^(b#5,3))P(D)=0  (Equation 7-5)(D ^(a#6,1,1) +D ^(a#6,1,2) +D ^(a#6,1,3))X ₁(D)+(D ^(a#6,2,1) +D^(a#6,2,2) +D ^(a#6,2,3))X ₂(D)+ . . . +(D ^(a#6,n-1,1) +D ^(a#6,n-1,2)+D ^(a#6,n-1,3))X _(n-1)(D)+(D ^(b#6,1) +D ^(b#6,2) +D^(b#6,3))P(D)=0  (Equation 7-6)

At this time, X₁(D), X₂(D), . . . , X_(n-1)(D) are polynomialrepresentations of data (information) X1, X2, . . . , Xn−1, and P(D) isa polynomial representation of parity. Here, in equations 7-1 to 7-6,parity check polynomials are assumed such that there are three terms inX₁(D), X₂(D), . . . , X_(n-1)(D), and P(D) respectively. As in the caseof the above coding rate of ½, and in the case of a time varying periodof 3, the possibility of being able to obtain higher error correctioncapability is increased if the condition below (<Condition #1>) issatisfied in an LDPC-CC of a time varying period of 6 and a coding rateof (n−1)/n (where n is an integer equal to or greater than 2)represented by parity check polynomials of equations 7-1 to 7-6.

In an LDPC-CC of a time varying period of 6 and a coding rate of (n−1)/n(where n is an integer equal to or greater than 2), parity andinformation at time i are represented by Pi and X_(i,1), X_(i,2), . . ., X_(i,n-1) respectively. If i %6=k (where k=0, 1, 2, 3, 4, 5) isassumed at this time, a parity check polynomial of equation 7−(k+1)holds true. For example, if i=8, i %6=2 (k=2), and therefore equation 8holds true.[8](D ^(a#3,1,1) +D ^(a#3,1,2) +D ^(a#3,1,3))X _(8,1)+(D ^(a#3,2,1) +D^(a#3,2,2) +D ^(a#3,2,3))X _(8,2)+ . . . +(D ^(a#3,n-1,1) +D^(a#3,n-1,2) +D ^(a#3,n-1,3))X _(8,n-2)+(D ^(b#3,1) +D ^(b#3,2) +D^(b#3,3))P ₈=0  (Equation 8)

<Condition #1>

In equations 7-1 to 7-6, combinations of orders of X₁(D), X₂(D), . . . ,X_(n-1)(D), and P(D) satisfy the following condition:

(a_(#1,1,1)%3, a_(#1,1,2)%3, a_(#1,1,3)%3),

(a_(#1,2,1)%3, a_(#1,2,2)%3, a_(#1,2,3)%3), . . . ,

(a_(#1,k,1)%3, a_(#1,k,2)%3, a_(#1,k,3)%3), . . . ,

(a_(#1,n-1,1)%3, a_(#1,n-1,2)%3, a_(#1,n-1,3)%3), and

(b_(#1,1)%3, b_(#1,2)%3, b_(#1,3)%3), are any of (0, 1, 2), (0, 2, 1),(1, 0, 2), (1, 2, 0), (2, 0, 1), or (2, 1, 0) (where k=1, 2, 3, . . . ,n−1);

(a_(#2,1,1)%3, a_(#2,1,2)%3, a_(#2,1,3)%3),

(a_(#2,2,1)%3, a_(#2,2,2)%3, a_(#2,2,3)%3), . . . ,

(a_(#2,k,1)%3, a_(#2,k,2)%3, a_(#2,k,3)%3), . . . ,

(a_(#2,n-1,1)%3, a_(#2,n-1,2)%3, a_(#2,n-1,3)%3), and

(b_(#2,1)%3, b_(#2,2)%3, b_(#2,3)%3), are any of (0, 1, 2), (0, 2, 1),(1, 0, 2), (1, 2, 0), (2, 0, 1), or (2, 1, 0) (where k=1, 2, 3, . . . ,n−1);

(a_(#3,1,1)%3, a_(#3,1,2)%3, a_(#3,1,3)%3),

(a_(#3,2,1)%3, a_(#3,2,2)%3, a_(#3,2,3)%3), . . . ,

(a_(#3,k,1)%3, a_(#3,k,2)%3, a_(#3,k,3)%3), . . . ,

(a_(#3,n-1,1)%3, a_(#3,n-1,2)%3, a_(#3,n-1,3)%3), and

(b_(#3,1)%3, b_(#3,2)%3, b_(#3,3)%3), are any of (0, 1, 2), (0, 2, 1),(1, 0, 2), (1, 2, 0), (2, 0, 1), or (2, 1, 0) (where k=1, 2, 3, . . . ,n−1);

(a_(#4,1,1)%3, a_(#4,1,2)%3, a_(#4,1,3)%3),

(a_(#4,2,1)%3, a_(#4,2,2)%3, a_(#4,2,3)%3), . . . ,

(a_(#4,k,1)%3, a_(#4,k,2)%3, a_(#4,k,3)%3), . . . ,

(a_(#4,n-1,1)%3, a_(#4,n-1,2)%3, a_(#4,n-1,3)%3), and

(b_(#4,1)%3, b_(#4,2)%3, b_(#4,3)%3), are any of (0, 1, 2), (0, 2, 1),(1, 0, 2), (1, 2, 0), (2, 0, 1), or (2, 1, 0) (where k=1, 2, 3, . . . ,n−1);

(a_(#5,1,1)%3, a_(#5,1,2)%3, a_(#5,1,3)%3),

(a_(#5,2,1)%3, a_(#5,2,2)%3, a_(#5,2,3)%3), . . . ,

(a_(#5,k,1)%3, a_(#5,k,2)%3, a_(#5,k,3)%3), . . . ,

(a_(#5,n-1,1)%3, a_(#5,n-1,2)%3, a_(#5,n-1,3)%3), and

(b_(#5,1)%3, b_(#5,2)%3, b_(#5,3)%3), are any of (0, 1, 2), (0, 2, 1),(1, 0, 2), (1, 2, 0), (2, 0, 1), or (2, 1, 0) (where k=1, 2, 3, . . . ,n−1);

(a_(#6,1,1)%3, a_(#6,1,2)%3, a_(#6,1,3)%3),

(a_(#6,2,1)%3, a_(#6,2,2)%3, a_(#6,2,3)%3), . . . ,

(a_(#6,k,1)%3, a_(#6,k,2)%3, a_(#6,k,3)%3), . . . ,

(a_(#6,n-1,1)%3, a_(#6,n-1,2)%3, a_(#6,n-1,3)%3), and

(b_(#6,1)%3, b_(#6,2)%3, b_(#6,3)%3), are any of (0, 1, 2), (0, 2, 1),(1, 0, 2), (1, 2, 0), (2, 0, 1), or (2, 1, 0) (where k=1, 2, 3, . . . ,n−1);

In the above description, a code having high error correction capabilityhas been described for an LDPC-CC of a time varying period of 6, but acode having high error correction capability can also be generated whenan LDPC-CC of a time varying period of 3g (where g=1, 2, 3, 4, . . . )(that is, an LDPC-CC for which the time varying period is a multiple of3) is created in the same way as with the design method for an LDPC-CCof a time varying period of 3 or 6. A configuration method for this codeis described in detail below.

Consider equations 9-1 to 9-3g as parity check polynomials of an LDPC-CCfor which the time varying period is 3g (where g=1, 2, 3, 4, . . . ) andthe coding rate is (n−1)/n (where n is an integer equal to or greaterthan 2).

$\begin{matrix}{\mspace{79mu}\lbrack 9\rbrack} & \; \\{{{( {D^{{a{\# 1}},1,1} + D^{{a{\# 1}},1,2} + D^{{a{\# 1}},1,3}} ){X_{1}(D)}} + {( {D^{{a{\# 1}},2,1} + D^{{a{\# 1}},2,2} + D^{{a{\# 1}},2,3}} ){X_{2}(D)}} + \ldots + {( {D^{{a{\# 1}},{n - 1},1} + D^{{a{\# 1}},{n - 1},2} + D^{{a{\# 1}},{n - 1},3}} ){X_{n - 1}(D)}} + {( {D^{{b{\# 1}},1} + D^{{b{\# 1}},2} + D^{{b{\# 1}},3}} ){P(D)}}} = 0} & ( {{Equation}\mspace{14mu} 9\text{-}1} ) \\{{{( {D^{{a{\# 2}},1,1} + D^{{a{\# 2}},1,2} + D^{{a{\# 2}},1,3}} ){X_{1}(D)}} + {( {D^{{a{\# 2}},2,1} + D^{{a{\# 2}},2,2} + D^{{a{\# 2}},2,3}} ){X_{2}(D)}} + \ldots + {( {D^{{a{\# 2}},{n - 1},1} + D^{{a{\# 2}},{n - 1},2} + D^{{a{\# 2}},{n - 1},3}} ){X_{n - 1}(D)}} + {( {D^{{b{\# 2}},1} + D^{{b{\# 2}},2} + D^{{b{\# 2}},3}} ){P(D)}}} = 0} & ( {{Equation}\mspace{14mu} 9\text{-}2} ) \\{{{( {D^{{a{\# 3}},1,1} + D^{{a{\# 3}},1,2} + D^{{a{\# 3}},1,3}} ){X_{1}(D)}} + {( {D^{{a{\# 3}},2,1} + D^{{a{\# 3}},2,2} + D^{{a{\# 3}},2,3}} ){X_{2}(D)}} + \ldots + {( {D^{{a{\# 3}},{n - 1},1} + D^{{a{\# 3}},{n - 1},2} + D^{{a{\# 3}},{n - 1},3}} ){X_{n - 1}(D)}} + {( {D^{{b{\# 3}},1} + D^{{b{\# 3}},2} + D^{{b{\# 3}},3}} ){P(D)}}} = 0} & ( {{Equation}\mspace{14mu} 9\text{-}3} ) \\{\mspace{79mu}\vdots\mspace{79mu}} & \; \\{{{( {D^{{a\# k},1,1} + D^{{a\# k},1,2} + D^{{a\# k},1,3}} ){X_{1}(D)}} + {( {D^{{a\# k},2,1} + D^{{a\# k},2,2} + D^{{a\# k},2,3}} ){X_{2}(D)}} + \ldots + {( {D^{{a\# k},{n - 1},1} + D^{{a\# k},{n - 1},2} + D^{{a\# k},{n - 1},3}} ){X_{n - 1}(D)}} + {( {D^{{b\# k},1} + D^{{b\# k},2} + D^{{b\# k},3}} ){P(D)}}} = 0} & ( {{Equation}\mspace{14mu} 9\text{-}k} ) \\{\mspace{79mu}\vdots} & \; \\{{{( {D^{{{a{\# 3}g} - 2},1,1} + D^{{{a{\# 3}g} - 2},1,2} + D^{{{a{\# 3}g} - 2},1,3}} ){X_{1}(D)}} + {( {D^{{{a{\# 3}g} - 2},2,1} + D^{{{a{\# 3}g} - 2},2,2} + D^{{{a{\# 3}g} - 2},2,3}} ){X_{2}(D)}} + \ldots + {( {D^{{{a{\# 3}g} - 2},{n - 1},1} + D^{{{a{\# 3}g} - 2},{n - 1},2} + D^{{{a{\# 3}g} - 2},{n - 1},3}} ){X_{n - 1}(D)}} + {( {D^{{{b{\# 3}g} - 2},1} + D^{{{b{\# 3}g} - 2},2} + D^{{{b{\# 3}g} - 2},3}} ){P(D)}}} = 0} & ( {{Equation}\mspace{14mu} 9\text{-}( {3g\text{-}2} )} ) \\{{{( {D^{{{a{\# 3}g} - 1},1,1} + D^{{{a{\# 3}g} - 1},1,2} + D^{{{a{\# 3}g} - 1},1,3}} ){X_{1}(D)}} + {( {D^{{{a{\# 3}g} - 1},2,1} + D^{{{a{\# 3}g} - 1},2,2} + D^{{{a{\# 3}g} - 1},2,3}} ){X_{2}(D)}} + \ldots + {( {D^{{{a{\# 3}g} - 1},{n - 1},1} + D^{{{a{\# 3}g} - 1},{n - 1},2} + D^{{{a{\# 3}g} - 1},{n - 1},3}} ){X_{n - 1}(D)}} + {( {D^{{{b{\# 3}g} - 1},1} + D^{{{b{\# 3}g} - 1},2} + D^{{{b{\# 3}g} - 1},3}} ){P(D)}}} = 0} & ( {{Equation}\mspace{14mu} 9\text{-}( {3g\text{-}1} )} ) \\{{{( {D^{{a{\# 3}g},1,1} + D^{{a{\# 3}g},1,2} + D^{{a{\# 3}g},1,3}} ){X_{1}(D)}} + {( {D^{{a{\# 3}g},2,1} + D^{{a{\# 3}g},2,2} + D^{{a{\# 3}g},2,3}} ){X_{2}(D)}} + \ldots + {( {D^{{a{\# 3}g},{n - 1},1} + D^{{a{\# 3}g},{n - 1},2} + D^{{a{\# 3}g},{n - 1},3}} ){X_{n - 1}(D)}} + {( {D^{{b{\# 3}g},1} + D^{{b{\# 3}g},2} + D^{{b{\# 3}g},3}} ){P(D)}}} = 0} & ( {{Equation}\mspace{14mu} 9\text{-}3g} )\end{matrix}$

At this time, X₁(D), X₂(D), . . . , X_(n-1)(D) are polynomialrepresentations of data (information) X₁, X₂, . . . , X_(n-1), and P(D)is a polynomial representation of parity. Here, in equations 9-1 to9-3g, parity check polynomials are assumed such that there are threeterms in X₁(D), X₂(D), . . . , X_(n-1)(D), and P(D) respectively.

As in the case of an LDPC-CC of a time varying period of 3 and anLDPC-CC of a time varying period of 6, the possibility of being able toobtain higher error correction capability is increased if the conditionbelow (<Condition #2>) is satisfied in an LDPC-CC of a time varyingperiod of 3g and a coding rate of (n−1)/n (where n is an integer equalto or greater than 2) represented by parity check polynomials ofequations 9-1 to 9-3g.

In an LDPC-CC of a time varying period of 3g and a coding rate of(n−1)/n (where n is an integer equal to or greater than 2), parity andinformation at time i are represented by P_(i) and X_(i,1), X_(i,2), . .. , X_(i,n-1) respectively. If i %3g=k (where k=0, 1, 2, . . . , 3g−1)is assumed at this time, a parity check polynomial of equation 9-(k+1)holds true. For example, if i=2, i %3g=2 (k=2), and therefore equation10 holds true.(D ^(a#3,1,1) +D ^(a#3,1,2) +D ^(a#3,1,3))X _(2,1)+(D ^(a#3,2,1) +D^(a#3,2,2) +D ^(a#3,2,3))X _(2,2)+ . . . +(D ^(a#3,n-1,1) +D^(a#3,n-1,2) +D ^(a#3,n-1,3))X _(2,n-1)+(D ^(b#3,1) +D ^(b#3,2) +D^(b#3,3))P ₂=0  (Equation 10)

In equations 9-1 to 9-3g, it is assumed that a_(#k,p,1), a_(#k,p,2) anda_(#k,p,3) are integers (where a_(#k,p,1)≠a_(#k,p,2)≠a_(#k,p,3)) (wherek=1, 2, 3, . . . , 3g, and p=1, 2, 3, . . . , n−1). Also, it is assumedthat b_(#k,1), b_(#k,2) and b_(#k,3) are integers (whereb_(#k,1)≠b_(#k,2)≠b_(k,3)). A parity check polynomial of equation 9-k(where k=1, 2, 3, . . . , 3g) is called “check equation #k,” and asub-matrix based on the parity check polynomial of equation 9-k isdesignated k-th sub-matrix H_(k). Next, an LDPC-CC of a time varyingperiod of 3g is considered that is generated from first sub-matrix H₁,second sub-matrix H₂, third sub-matrix H₃, . . . , and 3g-th sub-matrixH_(3g).

<Condition #2>

In equations 9-1 to 9-3g, combinations of orders of X₁(D), X₂(D), . . ., X_(n-1)(D), and

(a_(#1, 1, 1)%3, a_(#1, 1, 2)%3, a_(#1, 1, 3)%3), (a_(#1, 2, 1)%3, a_(#1, 2, 2)%3, a_(#1, 2, 3)%3), …  , (a_(#1, p, 1)%3, a_(#1, p, 2)%3, a_(#1, p, 3)%3), …  , (a_(#1, n − 1, 1)%3, a_(#1, n − 1, 2)%3, a_(#1, n − 1, 3)%3)  and(b_(#1, 1)%3, b_(#1, 2)%3, b_(#1, 3)%3)  are  any  of  (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), or  (2, 1, 0)(where  p = 1, 2, 3, …  ,,n − 1);(a_(#2, 1, 1)%3, a_(#2, 1, 2)%3, a_(#2, 1, 3)%3), (a_(#2, 2, 1)%3, a_(#2, 2, 2)%3, a_(#2, 2, 3)%3), …  , (a_(#2, p, 1)%3, a_(#2, p, 2)%3, a_(#2, p, 3)%3), …  , (a_(#2, n − 1, 1)%3, a_(#2, n − 1, 2)%3, a_(#2, n − 1, 3)%3)  and(b_(#2, 1)%3, b_(#2, 2)%3, b_(#2, 3)%3)  are  any  of  (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), or  (2, 1, 0)(where  p = 1, 2, 3, …  ,,n − 1);(a_(#3, 1, 1)%3, a_(#3, 1, 2)%3, a_(#3, 1, 3)%3), (a_(#3, 2, 1)%3, a_(#3, 2, 2)%3, a_(#3, 2, 3)%3), …  , (a_(#3, p, 1)%3, a_(#3, p, 2)%3, a_(#3, p, 3)%3), …  ,,(a_(#3, n − 1,)1%3, a_(#3, n − 1,)2%3, a_(#1, n − 1,)3%3)  and(b_(#3, 1)%3, b_(#3, 2)%3, b_(#3, 3)%3)  are  any  of  (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), or  (2, 1, 0)(where  p = 1, 2, 3, …  ,,n − 1);⋮(a_(#k, 1, 1)%3, a_(#k, 1, 2)%3, a_(#k, 1, 3)%3), (a_(#k, 2, 1)%3, a_(#k, 2, 2)%3, a_(#k, 2, 3)%3), …  , (a_(#k, p, 1)%3, a_(#k, p, 2)%3, a_(#k, p, 3)%3), …  , (a_(#k, n − 1, 1)%3, a_(#k, n − 1, 2)%3, a_(#k, n − 1, 3)%3)  and(b_(#k, 1)%3, b_(#k, 2)%3, b_(#k, 3)%3)  are  any  of  (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), or  (2, 1, 0)(where  p = 1, 2, 3, …  ,,n − 1)  (where, k = 1, 2, 3, …  , 3g);⋮(a_(#3g − 2, 1, 1)%3, a_(#3g − 2, 1, 2)%3, a_(#3g − 2, 1, 3)%3), (a_(#3g − 2, 2, 1)%3, a_(#3g − 2, 2, 2)%3, a_(#3g − 2, 2, 3)%3), …  , (a_(#3g − 2, p, 1)%3, a_(#3g − 2, p, 2)%3, a_(#3g − 2, p, 3)%3), …  , (a_(#3g − 2, n − 1, 1)%3, a_(#3g − 2, n − 1, 2)%3, a_(#3g − 2, n − 1, 3)%3), and(b_(#3g − 2, 1)%3, b_(#3g − 2, 2)%3, b_(#3g − 2, 3)%3)  are  any  of  (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), or  (2, 1, 0)(where  p = 1, 2, 3, …  ,,n − 1);(a_(#3g − 1, 1, 1)%3, a_(#3g − 1, 1, 2)%3, a_(#3g − 1, 1, 3)%3), (a_(#3g − 1, 2, 1)%3, a_(#3g − 1, 2, 2)%3, a_(#3g − 1, 2, 3)%3), …  , (a_(#3g − 1, p, 1)%3, a_(#3g − 1, p, 2)%3, a_(#3g − 1, p, 3)%3), …  , (a_(#3g − 1, n − 1, 1)%3, a_(#3g − 1, n − 1, 2)%3, a_(#3g − 1, n − 1, 3)%3)  and(b_(#3g − 1, 1)%3, b_(#3g − 1, 2)%3, b_(#3g − 1, 3)%3)  are  any  of  (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), or  (2, 1, 0)(where  p = 1, 2, 3, …  ,,n − 1); and(a_(#3g, 1, 1)%3, a_(#3g, 1, 2)%3, a_(#3g, 1, 3)%3), (a_(#3g, 2, 1)%3, a_(#3g, 2, 2)%3, a_(#3g, 2, 3)%3), …  , (a_(#3g, p, 1)%3, a_(#3g, p, 2)%3, a_(#3g, p, 3)%3), …  , (a_(#3g, n − 1, 1)%3, a_(#3g, n − 1, 2)%3, a_(#3g, n − 1, 3)%3)  and(b_(#3g, 1)%3, b_(#3g, 2)%3, b_(#3g, 3)%3)  are  any  of  (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), or  (2, 1, 0)(where  p = 1, 2, 3, …  ,,n − 1).

Taking ease of performing encoding into consideration, it is desirablefor one “0” to be present among the three items (b_(#k,1)%3, b#_(k,2)%3,b_(#k,3)%3) (where k=1, 2, . . . , 3g) in equations 9-1 to 9-3g. This isbecause of a feature that, if D⁰=1 holds true and b_(#k,1), b_(#k,2) andb_(#k,3) are integers equal to or greater than 0 at this time, parity Pcan be found sequentially.

Also, in order to provide relevancy between parity bits and data bits ofthe same point in time, and to facilitate a search for a code havinghigh correction capability, it is desirable for:

one “0” to be present among the three items (a_(#k,1,1)%3, a_(#k,1,2)%3,a_(#k,1,3)%3);

one “0” to be present among the three items (a_(#k,2,1)%3, a_(#k,2,2)%3,a_(#k,2,3)%3);

-   -   . . .

one “0” to be present among the three items (a_(#k,p,1)%3, a_(#k,p,2)%3,a_(#k,p,3)%3);

-   -   . . .

one “0” to be present among the three items (a_(#k,n-1,1)%3,a_(#k, n-1,2)%3, a_(#k, n-1,3)%3), (where k=1, 2, . . . , 3g).

Next, an LDPC-CC of a time varying period of 3g (where g=2, 3, 4, 5, . .. ) that takes ease of encoding into account is considered. At thistime, if the coding rate is (n−1)/n (where n is an integer equal to orgreater than 2), LDPC-CC parity check polynomials can be represented asshown below.

$\begin{matrix}{\mspace{79mu}\lbrack 11\rbrack} & \; \\{{{( {D^{{a{\# 1}},1,1} + D^{{a{\# 1}},1,2} + D^{{a{\# 1}},1,3}} ){X_{1}(D)}} + {( {D^{{a{\# 1}},2,1} + D^{{a{\# 1}},2,2} + D^{{a{\# 1}},2,3}} ){X_{2}(D)}} + \ldots + {( {D^{{a{\# 1}},{n - 1},1} + D^{{a{\# 1}},{n - 1},2} + D^{{a{\# 1}},{n - 1},3}} ){X_{n - 1}(D)}} + {( {D^{{b{\# 1}},1} + D^{{b{\# 1}},2} + D^{{b{\# 1}},3}} ){P(D)}}} = 0} & ( {{Equation}\mspace{14mu} 11\text{-}1} ) \\{{{( {D^{{a{\# 2}},1,1} + D^{{a{\# 2}},1,2} + D^{{a{\# 2}},1,3}} ){X_{1}(D)}} + {( {D^{{a{\# 2}},2,1} + D^{{a{\# 2}},2,2} + D^{{a{\# 2}},2,3}} ){X_{2}(D)}} + \ldots + {( {D^{{a{\# 2}},{n - 1},1} + D^{{a{\# 2}},{n - 1},2} + D^{{a{\# 2}},{n - 1},3}} ){X_{n - 1}(D)}} + {( {D^{{b{\# 2}},1} + D^{{b{\# 2}},2} + D^{{b{\# 2}},3}} ){P(D)}}} = 0} & ( {{Equation}\mspace{14mu} 11\text{-}2} ) \\{{{( {D^{{a{\# 3}},1,1} + D^{{a{\# 3}},1,2} + D^{{a{\# 3}},1,3}} ){X_{1}(D)}} + {( {D^{{a{\# 3}},2,1} + D^{{a{\# 3}},2,2} + D^{{a{\# 3}},2,3}} ){X_{2}(D)}} + \ldots + {( {D^{{a{\# 3}},{n - 1},1} + D^{{a{\# 3}},{n - 1},2} + D^{{a{\# 3}},{n - 1},3}} ){X_{n - 1}(D)}} + {( {D^{{b{\# 3}},1} + D^{{b{\# 3}},2} + D^{{b{\# 3}},3}} ){P(D)}}} = 0} & ( {{Equation}\mspace{14mu} 11\text{-}3} ) \\{\mspace{79mu}\vdots} & \; \\{{{( {D^{{a\# k},1,1} + D^{{a\# k},1,2} + D^{{a\# k},1,3}} ){X_{1}(D)}} + {( {D^{{a\# k},2,1} + D^{{a\# k},2,2} + D^{{a\# k},2,3}} ){X_{2}(D)}} + \ldots + {( {D^{{a\# k},{n - 1},1} + D^{{a\# k},{n - 1},2} + D^{{a\# k},{n - 1},3}} ){X_{n - 1}(D)}} + {( {D^{{b\# k},1} + D^{{b\# k},2} + D^{{b\# k},3}} ){P(D)}}} = 0} & ( {{Equation}\mspace{14mu} 11\text{-}k} ) \\{\mspace{79mu}\vdots} & \; \\{{{( {D^{{{a{\# 3}g} - 2},1,1} + D^{{{a{\# 3}g} - 2},1,2} + D^{{{a{\# 3}g} - 2},1,3}} ){X_{1}(D)}} + {( {D^{{{a{\# 3}g} - 2},2,1} + D^{{{a{\# 3}g} - 2},2,2} + D^{{{a{\# 3}g} - 2},2,3}} ){X_{2}(D)}} + \ldots + {( {D^{{{a{\# 3}g} - 2},{n - 1},1} + D^{{{a{\# 3}g} - 2},{n - 1},2} + D^{{{a{\# 3}g} - 2},{n - 1},3}} ){X_{n - 1}(D)}} + {( {D^{{{b{\# 3}g} - 2},1} + D^{{{b{\# 3}g} - 2},2} + D^{{{b{\# 3}g} - 2},3}} ){P(D)}}} = 0} & ( {{Equation}\mspace{14mu} 11\text{-}( {3g\text{-}2} )} ) \\{{{( {D^{{{a{\# 3}g} - 1},1,1} + D^{{{a{\# 3}g} - 1},1,2} + D^{{{a{\# 3}g} - 1},1,3}} ){X_{1}(D)}} + {( {D^{{{a{\# 3}g} - 1},2,1} + D^{{{a{\# 3}g} - 1},2,2} + D^{{{a{\# 3}g} - 1},2,3}} ){X_{2}(D)}} + \ldots + {( {D^{{{a{\# 3}g} - 1},{n - 1},1} + D^{{{a{\# 3}g} - 1},{n - 1},2} + D^{{{a{\# 3}g} - 1},{n - 1},3}} ){X_{n - 1}(D)}} + {( {D^{{{b{\# 3}g} - 1},1} + D^{{{b{\# 3}g} - 1},2} + D^{{{b{\# 3}g} - 1},3}} ){P(D)}}} = 0} & ( {{Equation}\mspace{14mu} 11\text{-}( {3g\text{-}1} )} ) \\{{{( {D^{{a{\# 3}g},1,1} + D^{{a{\# 3}g},1,2} + D^{{a{\# 3}g},1,3}} ){X_{1}(D)}} + {( {D^{{a{\# 3}g},2,1} + D^{{a{\# 3}g},2,2} + D^{{a{\# 3}g},2,3}} ){X_{2}(D)}} + \ldots + {( {D^{{a{\# 3}g},{n - 1},1} + D^{{a{\# 3}g},{n - 1},2} + D^{{a{\# 3}g},{n - 1},3}} ){X_{n - 1}(D)}} + {( {D^{{b{\# 3}g},1} + D^{{b{\# 3}g},2} + D^{{b{\# 3}g},3}} ){P(D)}}} = 0} & ( {{Equation}\mspace{14mu} 11\text{-}3g} )\end{matrix}$

At this time, X₁(D), X₂(D), . . . , X_(n-1)(D) are polynomialrepresentations of data (information) X₁, X₂, . . . , X_(n-1), and P(D)is a polynomial representation of parity. Here, in equations 11-1 to11-3g, parity check polynomials are assumed such that there are threeterms in X₁(D), X₂(D), . . . , X_(n-1)(D), and P(D) respectively. In anLDPC-CC of a time varying period of 3g and a coding rate of (n−1)/n(where n is an integer equal to or greater than 2), parity andinformation at time i are represented by Pi and X_(i,1), X_(i,2), . . ., X_(i,n-1) respectively. If i %3g=k (where k=0, 1, 2, . . . , 3g−1) isassumed at this time, a parity check polynomial of equation 11-(k+1)holds true. For example, if i=2, i %3=2 (k=2), and therefore equation 12holds true.(D ^(a#3,1,1) +D ^(a#3,1,2) +D ^(a#3,1,3))X _(2,1)+(D ^(a#3,2,1) +D^(a#3,2,2) +D ^(a#3,2,3))X _(2,2)+ . . . +(D ^(a#3,n-1,1) +D^(a#3,n-1,2) +D ^(a#3,n-1,3))X _(2,n-1)+(D ^(b#3,1) +D ^(b#3,2)+1)P₂=0  (Equation 12)

If <Condition #3> and <Condition #4> are satisfied at this time, thepossibility of being able to create a code having higher errorcorrection capability is increased.

<Condition #3>

In equations 11-1 to 11-3g, combinations of orders of X₁(D), X₂(D), . .. , X_(n-1)(D), and P(D) satisfy the following condition:

(a_(#1, 1, 1)%3, a_(#1, 1, 2)%3, a_(#1, 1, 3)%3), (a_(#1, 2, 1)%3, a_(#1, 2, 2)%3, a_(#1, 2, 3)%3), …  , (a_(#1, p, 1)%3, a_(#1, p, 2)%3, a_(#1, p, 3)%3), …  , and(a_(#1, n − 1, 1)%3, a_(#1, n − 1, 2)%3, a_(#1, n − 1, 3)%3)  are  any  of  (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), or  (2, 1, 0)(where  p = 1, 2, 3, …  , n − 1);(a_(#2, 1, 1)%3, a_(#2, 1, 2)%3, a_(#2, 1, 3)%3), (a_(#2, 2, 1)%3, a_(#2, 2, 2)%3, a_(#2, 2, 3)%3), …  , (a_(#2, p, 1)%3, a_(#2, p, 2)%3, a_(#2, p, 3)%3), …  , and(a_(#2, n − 1, 1)%3, a_(#2, n − 1, 2)%3, a_(#2, n − 1, 3)%3)  are  any  of  (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), or  (2, 1, 0)(where  p = 1, 2, 3, …  , n − 1);(a_(#3, 1, 1)%3, a_(#3, 1, 2)%3, a_(#3, 1, 3)%3), (a_(#3, 2, 1)%3, a_(#3, 2, 2)%3, a_(#3, 2, 3)%3), …  , (a_(#3, p, 1)%3, a_(#3, p, 2)%3, a_(#3, p, 3)%3), …  , and(a_(#3, n − 1, 1)%3, a_(#3, n − 1, 2)%3, a_(#3, n − 1, 3)%3)  are  any  of  (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), or  (2, 1, 0)(where  p = 1, 2, 3, …  , n − 1);⋮(a_(#k, 1, 1)%3, a_(#k, 1, 2)%3, a_(#k, 1, 3)%3), (a_(#k, 2, 1)%3, a_(#k, 2, 2)%3, a_(#k, 2, 3)%3), …  , (a_(#k, p, 1)%3, a_(#k, p, 2)%3, a_(#k, p, 3)%3), …  , and(a_(#k, n − 1, 1)%3, a_(#k, n − 1, 2)%3, a_(#k, n − 1, 3)%3)  are  any  of  (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), or  (2, 1, 0)  (where  p = 1, 2, 3, …  , n − 1, and  k = 1, 2, 3, …  , 3g);⋮(a_(#3g − 2, 1, 1)%3, a_(#3g − 2, 1, 2)%3, a_(#3g − 2, 1, 3)%3), (a_(#3g − 2, 2, 1)%3, a_(#3g − 2, 2, 2)%3, a_(#3g − 2, 2, 3)%3), …  , (a_(#3g − 2, p, 1)%3, a_(#3g − 2, p, 2)%3, a_(#3g − 2, p, 3)%3), …  , and(a_(#3g − 2, n − 1, 1)%3, a_(#3g − 2, n − 1, 2)%3, a_(#3g − 2, n − 1, 3)%3)  are  any  of(0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), or  (2, 1, 0)(where  p = 1, 2, 3, …  , n − 1);(a_(#3g − 1, 1, 1)%3, a_(#3g − 1, 1, 2)%3, a_(#3g − 1, 1, 3)%3), (a_(#3g − 1, 2, 1)%3, a_(#3g − 1, 2, 2)%3, a_(#3g − 1, 2, 3)%3), …  , (a_(#3g − 1, p, 1)%3, a_(#3g − 1, p, 2)%3, a_(#3g − 1, p, 3)%3), …  , and(a_(#3g − 1, n − 1, 1)%3, a_(#3g − 1, n − 1, 2)%3, a_(#3g − 1, n − 1, 3)%3)  are  any  of(0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), or  (2, 1, 0)(where  p = 1, 2, 3, …  , n − 1); and(a_(#3g, 1, 1)%3, a_(#3g, 1, 2)%3, a_(#3g, 1, 3)%3), (a_(#3g, 2, 1)%3, a_(#3g, 2, 2)%3, a_(#3g, 2, 3)%3), …  , (a_(#3g, p, 1)%3, a_(#3g, p, 2)%3, a_(#3g, p, 3)%3), …  , and(a_(#3g, n − 1, 1)%3, a_(#3g, n − 1, 2)%3, a_(#3g, n − 1, 3)%3)  are  any  of  (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), or  (2, 1, 0)(where  p = 1, 2, 3, …  , n − 1).

In addition, in equations 11-1 to 11-3g, combinations of orders of P(D)satisfy the following condition:

(b#_(1,1)%3, b_(#1,2)%3),

(b_(#2,1)%3, b_(#2,2)%3),

(b_(#3,1)%3, b_(#3,2)%3), . . . ,

(b_(#k,1)%3, b_(#k,2)%3), . . . ,

(b_(#3g-2,1)%3, b_(#3g-2,2)%3),

(b_(#3g-1,1)%3, b_(#3g-1,2)%3), and

(b_(#3g,1)%3, b_(#3g,2)%3) are any of (1, 2), or (2, 1) (where k=1, 2,3, . . . , 3g).

<Condition #3> has a similar relationship with respect to equations 11-1to 11-3g as <Condition #2> has with respect to equations 9-1 to 9-3g. Ifthe condition below (<Condition #4>) is added for equations 11-1 to11-3g in addition to <Condition #3>, the possibility of being able tocreate an LDPC-CC having higher error correction capability isincreased.

<Condition #4>

Orders of P(D) of equations 11-1 to 11-3g satisfy the followingcondition:

all values other than multiples of 3 (that is, 0, 3, 6, . . . , 3g−3)from among integers from 0 to 3g−1 (0, 1, 2, 3, 4, . . . , 3g−2, 3g−1)are present in the values of 6g orders of

(b_(#1,1)%3g, b_(#1,2)%3g),

(b_(#2,1)%3g, b_(#2,2)%3g),

(b_(#3,1)%3g, b_(#3,2)%3g), . . . ,

(b_(#k,1)%3g, b_(#k,2)%3g), . . . ,

(b_(#3g-2,1)%3g, b_(#3g-2,2)%3g),

(b_(#3g-1,1)%3g, b_(#3g-1,2)%3g),

(b_(#3g,1)%3g, b_(#3g,2)%3g) (in this case, two orders form a pair, andtherefore the number of orders forming 3g pairs is 6g).

The possibility of obtaining good error correction capability is high ifthere is also randomness while regularity is maintained for positions atwhich “1”s are present in a parity check matrix. With an LDPC-CC forwhich the time varying period is 3g (where g=2, 3, 4, 5, . . . ) and thecoding rate is (n−1)/n (where n is an integer equal to or greater than2) that has parity check polynomials of equations 11-1 to 11-3g, if acode is created in which <Condition #4> is applied in addition to<Condition #3>, it is possible to provide randomness while maintainingregularity for positions at which “1”s are present in a parity checkmatrix, and therefore the possibility of obtaining good error correctioncapability is increased.

Next, an LDPC-CC of a time varying period of 3g (where g=2, 3, 4, 5, . .. ) is considered that enables encoding to be performed easily andprovides relevancy to parity bits and data bits of the same point intime. At this time, if the coding rate is (n−1)/n (where n is an integerequal to or greater than 2), LDPC-CC parity check polynomials can berepresented as shown below.

$\begin{matrix}{\mspace{79mu}\lbrack 13\rbrack} & \; \\{{{( {D^{{a{\# 1}},1,1} + D^{{a{\# 1}},1,2} + D^{{a{\# 1}},1,3}} ){X_{1}(D)}} + {( {D^{{a{\# 1}},2,1} + D^{{a{\# 1}},2,2} + D^{{a{\# 1}},2,3}} ){X_{2}(D)}} + \ldots + {( {D^{{a{\# 1}},{n - 1},1} + D^{{a{\# 1}},{n - 1},2} + D^{{a{\# 1}},{n - 1},3}} ){X_{n - 1}(D)}} + {( {D^{{b{\# 1}},1} + D^{{b{\# 1}},2} + D^{{b{\# 1}},3}} ){P(D)}}} = 0} & ( {{Equation}\mspace{14mu} 13\text{-}1} ) \\{{{( {D^{{a{\# 2}},1,1} + D^{{a{\# 2}},1,2} + D^{{a{\# 2}},1,3}} ){X_{1}(D)}} + {( {D^{{a{\# 2}},2,1} + D^{{a{\# 2}},2,2} + D^{{a{\# 2}},2,3}} ){X_{2}(D)}} + \ldots + {( {D^{{a{\# 2}},{n - 1},1} + D^{{a{\# 2}},{n - 1},2} + D^{{a{\# 2}},{n - 1},3}} ){X_{n - 1}(D)}} + {( {D^{{b{\# 2}},1} + D^{{b{\# 2}},2} + D^{{b{\# 2}},3}} ){P(D)}}} = 0} & ( {{Equation}\mspace{14mu} 13\text{-}2} ) \\{{{( {D^{{a{\# 3}},1,1} + D^{{a{\# 3}},1,2} + D^{{a{\# 3}},1,3}} ){X_{1}(D)}} + {( {D^{{a{\# 3}},2,1} + D^{{a{\# 3}},2,2} + D^{{a{\# 3}},2,3}} ){X_{2}(D)}} + \ldots + {( {D^{{a{\# 3}},{n - 1},1} + D^{{a{\# 3}},{n - 1},2} + D^{{a{\# 3}},{n - 1},3}} ){X_{n - 1}(D)}} + {( {D^{{b{\# 3}},1} + D^{{b{\# 3}},2} + D^{{b{\# 3}},3}} ){P(D)}}} = 0} & ( {{Equation}\mspace{14mu} 13\text{-}3} ) \\{\mspace{79mu}\vdots} & \; \\{{{( {D^{{a\# k},1,1} + D^{{a\# k},1,2} + D^{{a\# k},1,3}} ){X_{1}(D)}} + {( {D^{{a\# k},2,1} + D^{{a\# k},2,2} + D^{{a\# k},2,3}} ){X_{2}(D)}} + \ldots + {( {D^{{a\# k},{n - 1},1} + D^{{a\# k},{n - 1},2} + D^{{a\# k},{n - 1},3}} ){X_{n - 1}(D)}} + {( {D^{{b\# k},1} + D^{{b\# k},2} + D^{{b\# k},3}} ){P(D)}}} = 0} & ( {{Equation}\mspace{14mu} 13\text{-}k} ) \\{\mspace{79mu}\vdots} & \; \\{{{( {D^{{{a{\# 3}g} - 2},1,1} + D^{{{a{\# 3}g} - 2},1,2} + D^{{{a{\# 3}g} - 2},1,3}} ){X_{1}(D)}} + {( {D^{{{a{\# 3}g} - 2},2,1} + D^{{{a{\# 3}g} - 2},2,2} + D^{{{a{\# 3}g} - 2},2,3}} ){X_{2}(D)}} + \ldots + {( {D^{{{a{\# 3}g} - 2},{n - 1},1} + D^{{{a{\# 3}g} - 2},{n - 1},2} + D^{{{a{\# 3}g} - 2},{n - 1},3}} ){X_{n - 1}(D)}} + {( {D^{{{b{\# 3}g} - 2},1} + D^{{{b{\# 3}g} - 2},2} + D^{{{b{\# 3}g} - 2},3}} ){P(D)}}} = 0} & ( {{Equation}\mspace{14mu} 13\text{-}( {3g\text{-}2} )} ) \\{{{( {D^{{{a{\# 3}g} - 1},1,1} + D^{{{a{\# 3}g} - 1},1,2} + D^{{{a{\# 3}g} - 1},1,3}} ){X_{1}(D)}} + {( {D^{{{a{\# 3}g} - 1},2,1} + D^{{{a{\# 3}g} - 1},2,2} + D^{{{a{\# 3}g} - 1},2,3}} ){X_{2}(D)}} + \ldots + {( {D^{{{a{\# 3}g} - 1},{n - 1},1} + D^{{{a{\# 3}g} - 1},{n - 1},2} + D^{{{a{\# 3}g} - 1},{n - 1},3}} ){X_{n - 1}(D)}} + {( {D^{{{b{\# 3}g} - 1},1} + D^{{{b{\# 3}g} - 1},2} + D^{{{b{\# 3}g} - 1},3}} ){P(D)}}} = 0} & ( {{Equation}\mspace{14mu} 13\text{-}( {3g\text{-}1} )} ) \\{{{( {D^{{a{\# 3}g},1,1} + D^{{a{\# 3}g},1,2} + D^{{a{\# 3}g},1,3}} ){X_{1}(D)}} + {( {D^{{a{\# 3}g},2,1} + D^{{a{\# 3}g},2,2} + D^{{a{\# 3}g},2,3}} ){X_{2}(D)}} + \ldots + {( {D^{{a{\# 3}g},{n - 1},1} + D^{{a{\# 3}g},{n - 1},2} + D^{{a{\# 3}g},{n - 1},3}} ){X_{n - 1}(D)}} + {( {D^{{b{\# 3}g},1} + D^{{b{\# 3}g},2} + D^{{b{\# 3}g},3}} ){P(D)}}} = 0} & ( {{Equation}\mspace{14mu} 13\text{-}g} )\end{matrix}$

At this time, X₁(D), X₂(D), . . . , X_(n-1)(D) are polynomialrepresentations of data (information) X₁, X₂, . . . , X_(n-1), and P(D)is a polynomial representation of parity. In equations 13-1 to 13-3g,parity check polynomials are assumed such that there are three terms inX₁(D), X₂(D), . . . , X_(n-1)(D), and P(D) respectively, and term D⁰ ispresent in X₁(D), X₂(D), . . . , X_(n-1)(D), and P(D) (where k=1, 2, 3,. . . , 3g).

In an LDPC-CC of a time varying period of 3g and a coding rate of(n−1)/n (where n is an integer equal to or greater than 2), parity andinformation at time i are represented by Pi and X_(i,1), X_(i,2), . . ., X_(i,n-1) respectively. If i %3g=k (where k=0, 1, 2, . . . 3g−1) isassumed at this time, a parity check polynomial of equation 13-(k+1)holds true. For example, if i=2, i %3g=2 (k=2), and therefore equation14 holds true.(D ^(a#3,1,1) +D ^(a#3,1,2)+1)X _(2,1)+(D ^(a#3,2,1) +D ^(a#3,2,2)1)X_(2,2)+ . . . +(D ^(a#3,n-1,1) +D ^(a#3,n-1,2)1)X _(2,n-1)+(D ^(b#3,1)+D ^(b#3,2)1)P ₂=0  (Equation 14)

If following <Condition #5> and <Condition #6> are satisfied at thistime, the possibility of being able to create a code having higher errorcorrection capability is increased.

<Condition #5>

In equations 13-1 to 13-3g, combinations of orders of X₁(D), X₂(D), . .. , X_(n-1)(D), and P(D) satisfy the following condition:

(a_(#1, 1, 1)%3, a_(#1, 1, 2)%3), (a_(#1, 2, 1)%3, a_(#1, 2, 2)%3), …  , (a_(#1, p, 1)%3, a_(#1, p, 2)%3), …  , and(a_(#1, n − 1, 1)%3, a_(#1, n − 1, 2)%3)  are  any  of  (1, 2), (2, 1)(p = 1, 2, 3, …  , n − 1);(a_(#2, 1, 1)%3, a_(#2, 1, 2)%3), (a_(#2, 2, 1)%3, a_(#2, 2, 2)%3), …  , (a_(#2, p, 1)%3, a_(#2, p, 2)%3), …  , and(a_(#2, n − 1, 1)%3, a_(#2, n − 1, 2)%3)  are  any  of  (1, 2), or  (2, 1)(where  p = 1, 2, 3, …  , n − 1);(a_(#3, 1, 1)%3, a_(#3, 1, 2)%3), (a_(#3, 2, 1)%3, a_(#3, 2, 2)%3), …  , (a_(#3, p, 1)%3, a_(#3, p, 2)%3), …  , and(a_(#3, n − 1, 1)%3, a_(#3, n − 1, 2)%3)  are  any  of  (1, 2), or  (2, 1)(where  p = 1, 2, 3, …  , n − 1);⋮(a_(#k, 1, 1)%3, a_(#k, 1, 2)%3), (a_(#k, 2, 1)%3, a_(#k, 2, 2)%3), …  , (a_(#k, p, 1)%3, a_(#k, p, 2)%3), …  , and(a_(#k, n − 1, 1)%3, a_(#k, n − 1, 2)%3)  are  any  of  (1, 2), or  (2, 1) (where  p = 1, 2, 3, …  , n − 1)  (where  k = 1, 2, 3, …  , 3g)⋮(a_(#3g − 2, 1, 1)%3, a_(#3g − 2, 1, 2)%3), (a_(#3g − 2, 2, 1)%3, a_(#3g − 2, 2, 2)%3), …  , (a_(#3g − 2, p, 1)%3, a_(#3g − 2, p, 2)%3), …  , and(a_(#3g − 2, n − 1, 1)%3, a_(#3g − 2, n − 1, 2)%3)  are  any  of  (1, 2), or  (2, 1)(where  p = 1, 2, 3, …  , n − 1);(a_(#3g − 1, 1, 1)%3, a_(#3g − 1, 1, 2)%3), (a_(#3g − 1, 2, 1)%3, a_(#3g − 1, 2, 2)%3), …  , (a_(#3g − 1, p, 1)%3, a_(#3g − 1, p, 2)%3), …  , and(a_(#3g − 1, n − 1, 1)%3, a_(#3g − 1, n − 1, 2)%3)  are  any  of  (1, 2), or  (2, 1)(where  p = 1, 2, 3, …  , n − 1); and(a_(#3g, 1, 1)%3, a_(#3g, 1, 2)%3), (a_(#3g, 2, 1)%3, a_(#3g, 2, 2)%3), …  , (a_(#3g, p, 1)%3, a_(#3g, p, 2)%3), …  , and(a_(#3g, n − 1, 1)%3, a_(#3g, n − 1, 2)%3)  are  any  of  (1, 2), or  (2, 1)(where  p = 1, 2, 3, …  , n − 1).

In addition, in equations 13-1 to 13-3g, combinations of orders of P(D)satisfy the following condition:

(b_(#1,1)%3, b_(#1,2)%3),

(b_(#2,1)%3, b_(#2,2)%3),

(b_(#3,1)%3, b_(#3,2)%3), . . . ,

(b_(#k,1)%3, b_(#k,2)%3), . . . ,

(b_(#3g-2,1)%3, b_(#3g-2,2)%3),

(b_(#3g-1,1)%3, b_(#3g-1,2)%3), and

(b_(#3g,1)%3, b_(#3g,2)%3) are any of (1, 2), or (2, 1) (where k=1, 2,3, . . . , 3g).

<Condition #5> has a similar relationship with respect to equations 13-1to 13-3g as <Condition #2> has with respect to equations 9-1 to 9-3g. Ifthe condition below (<Condition #6>) is added for equations 13-1 to13-3g in addition to <Condition #5>, the possibility of being able tocreate a code having high error correction capability is increased.

<Condition #6>

Orders of X₁(D) of equations 13-1 to 13-3g satisfy the followingcondition: all values other than multiples of 3 (that is, 0, 3, 6, . . ., 3g−3) from among integers from 0 to 3g−1 (0, 1, 2, 3, 4, . . . , 3g−2,3g−1) are present in the following 6g values of

(a_(#1,1,1)%3g, a_(#1,1,2)%3g),

(a_(#2,1,1)%3g, a_(#2,1,2)%3g), . . . ,

(a_(#p,1,1)%3g, a_(#p,1,2)%3g), . . . , and

(a_(#3g,1,1)%3g, a_(#3g,1,2)%3g) (where p=1, 2, 3, . . . , 3g);

Orders of X₂(D) of equations 13-1 to 13-3g satisfy the followingcondition: all values other than multiples of 3 (that is, 0, 3, 6, . . ., 3g−3) from among integers from 0 to 3g−1 (0, 1, 2, 3, 4, . . . , 3g−2,3g−1) are present in the following 6g values of

(a_(#1,2,1)%3g, a_(#1,2,2)%3g),

(a_(#2,2,1)%3g, a_(#2,2,2)%3g), . . . ,

(a_(#p,2,1)%3g, a_(#p,2,2)%3g), . . . , and

(a_(#3g,2,1)%3g, a_(#3g,2,2)%3g) (where p=1, 2, 3, . . . , 3g);

Orders of X₃(D) of equations 13-1 to 13-3g satisfy the followingcondition: all values other than multiples of 3 (that is, 0, 3, 6, . . ., 3g−3) from among integers from 0 to 3g−1 (0, 1, 2, 3, 4, . . . , 3g−2,3g−1) are present in the following 6g values of

(a_(#1, 3, 1)%3g, a_(#1, 3, 2)%3g), (a_(#2, 3, 1)%3g, a_(#2, 3, 2)%3g), …  , (a_(#p, 3, 1)%3g, a_(#p, 3, 2)%3g), …  , and(a_(#3g, 3, 1)%3g, a_(#3g, 3, 2)%3g)  (where  p = 1, 2, 3, …  , 3g);⋮

Orders of X_(k)(D) of equations 13-1 to 13-3g satisfy the followingcondition: all values other than multiples of 3 (that is, 0, 3, 6, . . ., 3g−3) from among integers from 0 to 3g−1 (0, 1, 2, 3, 4, . . . , 3g−2,3g−1) are present in the following 6g values of

  (a_(#1, k, 1)%3g, a_(#1, k, 2)%3g),  (a_(#2, k, 1)%3g, a_(#2, k, 2)%3g), …  ,  (a_(#p, k, 1)%3g, a_(#p, k, 2)%3g), …  , and(a_(#3g, k,)1%3g, a_(#3g, k,)2%3g)  (where  p = 1, 2, 3, …  , 3g, and  k = 1, 2, 3, …  , n − 1);  ⋮

Orders of X_(n-1)(D) of equations 13-1 to 13-3g satisfy the followingcondition: all values other than multiples of 3 (that is, 0, 3, 6, . . ., 3g−3) from among integers from 0 to 3g−1 (0, 1, 2, 3, 4, . . . , 3g−2,3g−1) are present in the following 6g values of

(a_(#1,n-1,1)%3g, a_(#1, n-1,2)%3g),

(a_(#2, n-1,1)%3g, a_(#2, n-1,2)%3g), . . . ,

(a_(#p,n-1,1)%3g, a_(#p,n-1,2)%3g), . . . , and

(a_(#3g,n-1,1)%3g, a_(#3g,n-1,2)%3g) (where p=1, 2, 3, . . . , 3g); and

Orders of P(D) of equations 13-1 to 13-3g satisfy the followingcondition: all values other than multiples of 3 (that is, 0, 3, 6, . . ., 3g−3) from among integers from 0 to 3g−1 (0, 1, 2, 3, 4, . . . , 3g−2,3g−1) are present in the following 6g values of

(b_(#1,1)%3g, b_(#1,2)%3g),

(b_(#2,1)%3g, b_(#2,2)%3g),

(b_(#3,1)%3g, b_(#3,2)%3g), . . . ,

(b_(#k,1)%3g, b_(#k,2)%3g), . . . ,

(b_(#3g-2,1)%3g, b_(#3g-2,2)%3g),

(b_(#3g-1,1)%3g, b_(#3g-1,2)%3g),

(b_(#3g,1)%3g, b_(#3g,2)%3g) (where k=1, 2, 3, . . . , n−1).

The possibility of obtaining good error correction capability is high ifthere is also randomness while regularity is maintained for positions atwhich “1”s are present in a parity check matrix. With an LDPC-CC forwhich the time varying period is 3g (where g=2, 3, 4, 5, . . . ) and thecoding rate is (n−1)/n (where n is an integer equal to or greater than2) that has parity check polynomials of equations 13-1 to 13-3g, if acode is created in which <Condition #6> is applied in addition to<Condition #5>, it is possible to provide randomness while maintainingregularity for positions at which “1”s are present in a parity checkmatrix, and therefore the possibility of obtaining good error correctioncapability is increased.

The possibility of being able to create an LDPC-CC having higher errorcorrection capability is also increased if a code is created using<Condition #6′> instead of <Condition #6>, that is, using <Condition#6′> in addition to <Condition #5>.

<Condition #6′>

Orders of X₁(D) of equations 13-1 to 13-3g satisfy the followingcondition: all values other than multiples of 3 (that is, 0, 3, 6, . . ., 3g−3) from among integers from 0 to 3g−1 (0, 1, 2, 3, 4, . . . , 3g−2,3g−1) are present in the following 6g values of

(a_(#1,1,1)%3g, a_(#1,1,2)%3g),

(a_(#2,1,1)%3g, a_(#2,1,2)%3g), . . . ,

(a_(#p,1,1)%3g, a_(#p,1,2)%3g), . . . , and

(a_(#3g,1,1)%3g, a_(#3g,1,2)%3g) (where p=1, 2, 3, . . . , 3g);

Orders of X₂(D) of equations 13-1 to 13-3g satisfy the followingcondition: all values other than multiples of 3 (that is, 0, 3, 6, . . ., 3g−3) from among integers from 0 to 3g−1 (0, 1, 2, 3, 4, 3g−2, 3g−1)are present in the following 6g values of

(a_(#1,2,1)%3g, a_(#1,2,2)%3g),

(a_(#2,2,1)%3g, a_(#2,2,2)%3g), . . . ,

(a_(#p,2,1)%3g, a_(#p,2,2)%3g), . . . , and

(a_(#3g,2,1)%3g, a_(#3g,2,2)%3g) (where p=1, 2, 3, . . . , 3g);

Orders of X₃(D) of equations 13-1 to 13-3g satisfy the followingcondition: all values other than multiples of 3 (that is, 0, 3, 6, 3g−3)from among integers from 0 to 3g−1 (0, 1, 2, 3, 4, . . . , 3g−2, 3g−1)are present in the following 6g values of

(a_(#1, 3, 1)%3g, a_(#1, 3, 2)%3g), (a_(#2, 3, 1)%3g, a_(#2, 3, 2)%3g), …  , (a_(#p, 3, 1)%3g, a_(#p, 3, 2)%3g), …  , and(a_(#3g, 3, 1)%3g, a_(#3g, 3, 2)%3g)  (where  p = 1, 2, 3, …  , 3g);⋮

Orders of X_(k)(D) of equations 13-1 to 13-3g satisfy the followingcondition: all values other than multiples of 3 (that is, 0, 3, 6, . . ., 3g−3) from among integers from 0 to 3g−1 (0, 1, 2, 3, 4, . . . , 3g−2,3g−1) are present in the following 6g values of

  (a_(#1, k, 1)%3g, a_(#1, k, 2)%3g),  (a_(#2, k, 1)%3g, a_(#2, k, 2)%3g), …  ,  (a_(#p, k, 1)%3g, a_(#p, k, 2)%3g), …  , and(a_(#3g, k, 1)%3g, a_(#3g, k, 2)%3g)  (where  p = 1, 2, 3, …  , 3g, and  k = 1, 2, 3, …  , n − 1);  ⋮

Orders of X_(n-1)(D) of equations 13-1 to 13-3g satisfy the followingcondition: all values other than multiples of 3 (that is, 0, 3, 6, . . ., 3g−3) from among integers from 0 to 3g−1 (0, 1, 2, 3, 4, . . . , 3g−2,3g−1) are present in the following 6g values of

(a_(#1,n-1,1)%3g, a_(#1,n-1,2)%3g),

(a_(#2,n-1,1)%3g, a_(#2,n-1,2)%3g), . . . ,

(a_(#p,n-1,1)%3g, a_(#p,n-1,2)%3g), . . . ,

(a_(#3g,n-1,1)%3g, a_(#3g,n-1,2)%3g) (where p=1, 2, 3, . . . , 3g); or

Orders of P(D) of equations 13-1 to 13-3g satisfy the followingcondition: all values other than multiples of 3 (that is, 0, 3, 6, . . ., 3g−3) from among integers from 0 to 3g−1 (0, 1, 2, 3, 4, . . . , 3g−2,3g−1) are present in the following 6g values of

(b_(#1,1)%3g, b_(#1,2)%3g),

(b_(#2,1)%3g, b_(#2,2)%3g),

(b_(#3,1)%3g, b_(#3,2)%3g), . . . ,

(b_(#k,1)%3g, b_(#k,2)%3g), . . . ,

(b_(#3g-2,1)%3g, b_(#3g-2,2)%3g),

(b_(#3g-1,1)%3g, b_(#3g-1,2)%3g),

(b_(#3g,1)%3g, b_(#3g,2)%3g) (where k=1, 2, 3, . . . , 3g).

The above description relates to an LDPC-CC of a time varying period of3g and a coding rate of (n−1)/n (where n is an integer equal to orgreater than 2). Below, conditions are described for orders of anLDPC-CC of a time varying period of 3g and a coding rate of ½ (n=2).

Consider equations 15-1 to 15-3g as parity check polynomials of anLDPC-CC for which the time varying period is 3g (where g=1, 2, 3, 4, . .. ) and the coding rate is ½ (n=2).

$\begin{matrix}{\mspace{20mu}\lbrack 15\rbrack} & \; \\{{{( {D^{{a{\# 1}},1,1} + D^{{a{\# 1}},1,2} + D^{{a{\# 1}},1,3}} ){X(D)}} + {( {D^{{b{\# 1}},1} + D^{{b{\# 1}},2} + D^{{b{\# 1}},3}} ){P(D)}}} = 0} & ( {{Equation}\mspace{14mu} 15\text{-}1} ) \\{{{( {D^{{a{\# 2}},1,1} + D^{{a{\# 2}},1,2} + D^{{a{\# 2}},1,3}} ){X(D)}} + {( {D^{{b{\# 2}},1} + D^{{b{\# 2}},2} + D^{{b{\# 2}},3}} ){P(D)}}} = 0} & ( {{Equation}\mspace{14mu} 15\text{-}2} ) \\{{{( {D^{{a{\# 3}},1,1} + D^{{a{\# 3}},1,2} + D^{{a{\# 3}},1,3}} ){X(D)}} + {( {D^{{b{\# 3}},1} + D^{{b{\# 3}},2} + D^{{b{\# 3}},3}} ){P(D)}}} = 0} & ( {{Equation}\mspace{14mu} 15\text{-}3} ) \\{\mspace{20mu}\vdots} & \; \\{{{( {D^{{a\# k},1,1} + D^{{a\# k},1,2} + D^{{a\# k},1,3}} ){X(D)}} + {( {D^{{b\# k},1} + D^{{b\# k},2} + D^{{b\# k},3}} ){P(D)}}} = 0} & ( {{Equation}\mspace{14mu} 15\text{-}k} ) \\{\mspace{20mu}\vdots} & \; \\{{{( {D^{{{a{\# 3}g} - 2},1,1} + D^{{{a{\# 3}g} - 2},1,2} + D^{{{a{\# 3}g} - 2},1,3}} ){X(D)}} + {( {D^{{{b{\# 3}g} - 2},1} + D^{{{b{\# 3}g} - 2},2} + D^{{{b{\# 3}g} - 2},3}} ){P(D)}}} = 0} & ( {{Equation}\mspace{14mu} 15\text{-}( {{3g} - 2} )} ) \\{{{( {D^{{{a{\# 3}g} - 1},1,1} + D^{{{a{\# 3}g} - 1},1,2} + D^{{{a{\# 3}g} - 1},1,3}} ){X(D)}} + {( {D^{{{b{\# 3}g} - 1},1} + D^{{{b{\# 3}g} - 1},2} + D^{{{b{\# 3}g} - 1},3}} ){P(D)}}} = 0} & ( {{Equation}\mspace{14mu} 15\text{-}( {{3g} - 1} )} ) \\{{{( {D^{{a{\# 3}g},1,1} + D^{{a{\# 3}g},1,2} + D^{{a{\# 3}g},1,3}} ){X(D)}} + {( {D^{{b{\# 3}g},1} + D^{{b{\# 3}g},2} + D^{{b{\# 3}g},3}} ){P(D)}}} = 0} & ( {{Equation}\mspace{14mu} 15\text{-}3g} )\end{matrix}$

At this time, X(D) is a polynomial representation of data (information)X and P(D) is a polynomial representation of parity. Here, in equations15-1 to 15-3g, parity check polynomials are assumed such that there arethree terms in X(D) and P(D) respectively.

Thinking in the same way as in the case of an LDPC-CC of a time varyingperiod of 3 and an LDPC-CC of a time varying period of 6, thepossibility of being able to obtain higher error correction capabilityis increased if the condition below (<Condition #2-1>) is satisfied inan LDPC-CC of a time varying period of 3g and a coding rate of ½ (n=2)represented by parity check polynomials of equations 15-1 to 15-3g.

In an LDPC-CC of a time varying period of 3g and a coding rate of ½(n=2), parity and information at time i are represented by Pi andX_(i,1) respectively. If i %3g=k (where k=0, 1, 2, . . . , 3g−1) isassumed at this time, a parity check polynomial of equation 15-(k+1)holds true. For example, if i=2, i %3g=2 (k=2), and therefore equation16 holds true.[16](D ^(a#3,1,1) +D ^(a#3,1,2) +D ^(a#3,1,3))X _(2,1)+(D ^(b#3,1) +D^(b#3,2) +D ^(b#3,3))P ₂=0  (Equation 16)

In equations 15-1 to 15-3g, it is assumed that a_(#k,1,1), a_(#k,1,2),and a_(#k,1,3) are integers (where a_(#k,1,1)≠a_(#k,1,2)≠a_(#k,1,3))(where k=1, 2, 3, . . . , 3g). Also, it is assumed that b_(#k,1),b_(#k,2), and b_(#k,3) are integers (where b_(#k,1)≠b_(#k,2)≠b_(#k,3)).A parity check polynomial of equation 15-k (k=1, 2, 3, . . . , 3g) iscalled “check equation #k,” and a sub-matrix based on the parity checkpolynomial of equation 15-k is designated k-th sub-matrix H_(k). Next,an LDPC-CC of a time varying period of 3g is considered that isgenerated from first sub-matrix H₁, second sub-matrix H₂, thirdsub-matrix H₃, . . . , and 3g-th sub-matrix H_(3g).

<Condition #2-1>

In equations 15-1 to 15-3g, combinations of orders of X(D) and P(D)satisfy the following condition:

(a_(#1, 1, 1)%3, a_(#1, 1, 2)%3, a_(#1, 1, 3)%3)  and  (b_(#1, 1)%3, b_(#1, 2)%3, b_(#1, 3)%3)  are  any  of  (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), or  (2, 1, 0);(a_(#2, 1, 1)%3, a_(#2, 1, 2)%3, a_(#2, 1, 3)%3)  and  (b_(#2, 1)%3, b_(#2, 2)%3, b_(#2, 3)%3)  are  any  of  (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), or  (2, 1, 0);(a_(#3, 1, 1)%3, a_(#3, 1, 2)%3, a_(#3, 1, 3)%3)  and  (b_(#3, 1)%3, b_(#3, 2)%3, b_(#3, 3)%3)  are  any  of  (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), or  (2, 1, 0);  ⋮(a_(#k, 1, 1)%3, a_(#k, 1, 2)%3, a_(#k, 1, 3)%3)  and  (b_(#k, 1)%3, b_(#k, 2)%3, b_(#k, 3)%3)  are  any  of  (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), or  (2, 1, 0)  (where  k = 1, 2, 3, …  , 3g);  ⋮(a_(#3g − 2, 1, 1)%3, a_(#3g − 2, 1, 2)%3, a_(#3g − 2, 1, 3)%3)  and  (b_(#3g − 2, 1)%3, b_(#3g − 2, 2)%3, b_(#3g − 2, 3)%3)  are  any  of  (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), or  (2, 1, 0);(a_(#3g − 1, 1, 1)%3, a_(#3g − 1, 1, 2)%3, a_(#3g − 1, 1, 3)%3)  and  (b_(#3g − 1, 1)%3, b_(#3g − 1, 2)%3, b_(#3g − 1, 3)%3)  are  any  of  (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), or  (2, 1, 0); and(a_(#3g, 1, 1)%3, a_(#3g, 1, 2)%3, a_(#3g, 1, 3)%3)  and  (b_(#3g, 1)%3, b_(#3g, 2)%3, b_(#3g, 3)%3)  are  any  of  (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), or  (2, 1, 0).

Taking ease of performing encoding into consideration, it is desirablefor one “0” to be present among the three items (b_(#k,1)%3, b_(#k,2)%3,b_(#k,3)%3) (where k=1, 2, . . . , 3g) in equations 15-1 to 15-3g. Thisis because of a feature that, if D⁰=1 holds true and b_(#k,1), b_(#k,2)and b_(#k,3) are integers equal to or greater than 0 at this time,parity P can be found sequentially.

Also, in order to provide relevancy between parity bits and data bits ofthe same point in time, and to facilitate a search for a code havinghigh correction capability, it is desirable for one “0” to be presentamong the three items (a_(#k,1,1)%3, a_(#k,1,2)%3, a_(#k,1,3)%3) (wherek=1, 2, . . . , 3g).

Next, an LDPC-CC of a time varying period of 3g (where g=2, 3, 4, 5, . .. ) that takes ease of encoding into account is considered. At thistime, if the coding rate is ½ (n=2), LDPC-CC parity check polynomialscan be represented as shown below.[17](D ^(a#1,1,1) +D ^(a#1,1,2) +D ^(a#1,1,3))X(D)+(D ^(b#1,1) +D^(b#1,2)+1)P(D)=0  (Equation 17-1)(D ^(a#2,1,1) +D ^(a#2,1,2) +D ^(a#2,1,3))X(D)+(D ^(b#2,1) +D^(b#2,2)+1)P(D)=0  (Equation 17-2)(D ^(a#3,1,1) +D ^(a#3,1,2) +D ^(a#3,1,3))X(D)+(D ^(b#3,1) +D^(b#3,2)+1)P(D)=0  (Equation 17-3)

At this time, X(D) is a polynomial representation of data (information)X and P(D) is a polynomial representation of parity. Here, in equations17-1 to 17-3g, parity check polynomials are assumed such that there arethree terms in X(D) and P(D) respectively. In an LDPC-CC of a timevarying period of 3g and a coding rate of ½ (n=2), parity andinformation at time i are represented by Pi and X_(i,1) respectively. Ifi %3g=k (where k=0, 1, 2, . . . , 3g−1) is assumed at this time, aparity check polynomial of equation 17-(k+1) holds true. For example, ifi=2, i %3g=2 (k=2), and therefore equation 18 holds true.[18](D ^(a#3,1,1) +D ^(a#3,1,2) +D ^(a#3,1,3))X _(2,1)+(D ^(b#3,1) +D^(b#3,2)+1)P ₂=0  (Equation 18)

If <Condition #3-1> and <Condition #4-1> are satisfied at this time, thepossibility of being able to create a code having higher errorcorrection capability is increased.

<Condition #3-1>

In equations 17-1 to 17-3g, combinations of orders of X(D) satisfy thefollowing condition:

(a_(#1, 1, 1)%3, a_(#1, 1, 2)%3, a_(#1, 1, 3)%3)  and  (b_(#1, 1)%3, b_(#1, 2)%3, b_(#1, 3)%3)  are  any  of  (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), or  (2, 1, 0);(a_(#2, 1, 1)%3, a_(#2, 1, 2)%3, a_(#2, 1, 3)%3)  and  (b_(#2, 1)%3, b_(#2, 2)%3, b_(#2, 3)%3)  are  any  of  (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), or  (2, 1, 0);(a_(#3, 1, 1)%3, a_(#3, 1, 2)%3, a_(#3, 1, 3)%3)  and  (b_(#3, 1)%3, b_(#3, 2)%3, b_(#3, 3)%3)  are  any  of  (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), or  (2, 1, 0);  ⋮(a_(#k, 1, 1)%3, a_(#k, 1, 2)%3, a_(#k, 1, 3)%3)  and  (b_(#k, 1)%3, b_(#k, 2)%3, b_(#k, 3)%3)  are  any  of  (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), or  (2, 1, 0)  (where  k = 1, 2, 3, …  , 3g);  ⋮(a_(#3g − 2, 1, 1)%3, a_(#3g − 2, 1, 2)%3, a_(#3g − 2, 1, 3)%3)  and  (b_(#3g − 2, 1)%3, b_(#3g − 2, 2)%3, b_(#3g − 2, 3)%3)  are  any  of  (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), or  (2, 1, 0);(a_(#3g − 1, 1, 1)%3, a_(#3g − 1, 1, 2)%3, a_(#3g − 1, 1, 3)%3)  and  (b_(#3g − 1, 1)%3, b_(#3g − 1, 2)%3, b_(#3g − 1, 3)%3)  are  any  of  (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), or  (2, 1, 0); and(a_(#3g, 1, 1)%3, a_(#3g, 1, 2)%3, a_(#3g, 1, 3)%3)  and  (b_(#3g, 1)%3, b_(#3g, 2)%3, b_(#3g, 3)%3)  are  any  of  (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), or  (2, 1, 0).

In addition, in equations 17-1 to 17-3g, combinations of orders of P(D)satisfy the following condition:

(b_(#1,1)%3, b_(#1,2)%3),

(b_(#2,1)%3, b_(#2,2)%3),

(b_(#3,1)%3, b_(#3,2)%3), . . . ,

(b_(#k,1)%3, b_(#k,2)%3), . . . ,

(b_(#3g-2,1)%3, b_(#3g-2,2)%3),

(b_(#3g-1,1)%3, b_(#3g-1,2)%3),

(b_(#3g,1)%3, b_(#3g,2)%3) are any of (1, 2), or (2, 1) (k=1, 2, 3, . .. , 3g).

<Condition #3-1> has a similar relationship with respect to equations17-1 to 17-3g as <Condition #2-1> has with respect to equations 15-1 to15-3g. If the condition below (<Condition #4-1>) is added for equations17-1 to 17-3g in addition to <Condition #3-1>, the possibility of beingable to create an LDPC-CC having higher error correction capability isincreased.

<Condition #4-1>

Orders of P(D) of equations 17-1 to 17-3g satisfy the followingcondition: all values other than multiples of 3 (that is, 0, 3, 6, . . ., 3g−3) from among integers from 0 to 3g−1 (0, 1, 2, 3, 4, . . . , 3g−2,3g−1) are present in the following 6g values of

(b_(#1,1)%3g, b_(#1,2)%3g),

(b_(#2,1)%3g, b_(#2,2)%3g),

(b_(#3,1)%3g, b_(#3,2)%3g), . . . ,

(b_(#k,1)%3g, b_(#k,2)%3g), . . . ,

(b_(#3g-2,1)%3g, b_(#3g-2,2)%3g),

(b_(#3g-1,1)%3g, b_(#3g-1,2)%3g), and

(b_(#3g,1)%3g, b_(#3g,2)%3g).

The possibility of obtaining good error correction capability is high ifthere is also randomness while regularity is maintained for positions atwhich “1”s are present in a parity check matrix. With an LDPC-CC forwhich the time varying period is 3g (where g=2, 3, 4, 5, . . . ) and thecoding rate is ½ (n=2) that has parity check polynomials of equations17-1 to 17-3g, if a code is created in which <Condition #4-1> is appliedin addition to <Condition #3-1>, it is possible to provide randomnesswhile maintaining regularity for positions at which “1”s are present ina parity check matrix, and therefore the possibility of obtaining bettererror correction capability is increased.

Next, an LDPC-CC of a time varying period of 3g (where 3, 4, 5, . . . )is considered that enables encoding to be performed easily and providesrelevancy to parity bits and data bits of the same point in time. Atthis time, if the coding rate is ½ (n=2), LDPC-CC parity checkpolynomials can be represented as shown below.

$\begin{matrix}{\mspace{20mu}\lbrack 19\rbrack} & \; \\{{{( {D^{{a{\# 1}},1,1} + D^{{a{\# 1}},1,2} + 1} ){X(D)}} + {( {D^{{b{\# 1}},1} + D^{{b{\# 1}},2} + 1} ){P(D)}}} = 0} & ( {{Equation}\mspace{14mu} 19\text{-}1} ) \\{{{( {D^{{a{\# 2}},1,1} + D^{{a{\# 2}},1,2} + 1} ){X(D)}} + {( {D^{{b{\# 2}},1} + D^{{b{\# 2}},2} + 1} ){P(D)}}} = 0} & ( {{Equation}\mspace{14mu} 19\text{-}2} ) \\{{{( {D^{{a{\# 3}},1,1} + D^{{a{\# 3}},1,2} + 1} ){X(D)}} + {( {D^{{b{\# 3}},1} + D^{{b{\# 3}},2} + 1} ){P(D)}}} = 0} & ( {{Equation}\mspace{14mu} 19\text{-}3} ) \\{\mspace{20mu}\vdots} & \; \\{{{( {D^{{a\# k},1,1} + D^{{a\# k},1,2} + 1} ){X(D)}} + {( {D^{{b\# k},1} + D^{{b\# k},2} + 1} ){P(D)}}} = 0} & ( {{Equation}\mspace{14mu} 19\text{-}k} ) \\{\mspace{20mu}\vdots} & \; \\{{{( {D^{{{a{\# 3}g} - 2},1,1} + D^{{{a{\# 3}g} - 2},1,2} + 1} ){X(D)}} + {( {D^{{{b{\# 3}g} - 2},1} + D^{{{b{\# 3}g} - 2},2} + 1} ){P(D)}}} = 0} & ( {{Equation}\mspace{14mu} 19\text{-}( {{3g} - 2} )} ) \\{{{( {D^{{{a{\# 3}g} - 1},1,1} + D^{{{a{\# 3}g} - 1},1,2} + 1} ){X(D)}} + {( {D^{{{b{\# 3}g} - 1},1} + D^{{{b{\# 3}g} - 1},2} + 1} ){P(D)}}} = 0} & ( {{Equation}\mspace{14mu} 19\text{-}( {{3g} - 1} )} ) \\{{{( {D^{{a{\# 3}g},1,1} + D^{{a{\# 3}g},1,2} + 1} ){X(D)}} + {( {D^{{b{\# 3}g},1} + D^{{b{\# 3}g},2} + 1} ){P(D)}}} = 0} & ( {{Equation}\mspace{14mu} 19\text{-}3g} )\end{matrix}$

At this time, X(D) is a polynomial representation of data (information)X and P(D) is a polynomial representation of parity.

In equations 19-1 to 19-3g, parity check polynomials are assumed suchthat there are three terms in X(D) and P(D) respectively, and a D⁰ termis present in X(D) and P(D) (where k=1, 2, 3, . . . , 3g).

In an LDPC-CC of a time varying period of 3g and a coding rate of ½(n=2), parity and information at time i are represented by Pi andX_(i,1) respectively. If i %3g=k (where k=0, 1, 2, . . . , 3g−1) isassumed at this time, a parity check polynomial of equation 19-(k+1)holds true. For example, if i=2, i %3g=2 (k=2), and therefore equation20 holds true.[20](D ^(a#3,1,1) +D ^(a#3,1,2)+1)X _(2,1)+(D ^(b#3,1) +D ^(b#3,2)+1)P₂=0  (Equation 20)

If following <Condition #5-1> and <Condition #6-1> are satisfied at thistime, the possibility of being able to create a code having higher errorcorrection capability is increased.

<Condition #5-1>

In equations 19-1 to 19-3g, combinations of orders of X(D) satisfy thefollowing condition:

  (a_(#1, 1, 1)%3, a_(#1, 1, 2)%3)  is  (1, 2)  or  (2, 1);  (a_(#2, 1, 1)%3, a_(#2, 1, 2)%3)  is  (1, 2)  or  (2, 1);  (a_(#3, 1, 1)%3, a_(#3, 1, 2)%3)  is  (1, 2)  or  (2, 1);  ⋮(a_(#k, 1, 1)%3, a_(#k, 1, 2)%3)  is  (1, 2)  or  (2, 1)  (where  k = 1, 2, 3, …  , 3g);  ⋮  (a_(#3g − 2, 1, 1)%3, a_(#3g − 2, 1, 2)%3)  is  (1, 2)  or  (2, 1),  (a_(#3g − 1, 1, 1)%3, a_(#3g − 1, 1, 2)%3)  is  (1, 2)  or  (2, 1); and  (a_(#3g, 1, 1)%3, a_(#3g, 1, 2)%3)  is  (1, 2)  or  (2, 1).

In addition, in equations 19-1 to 19-3g, combinations of orders of P(D)satisfy the following condition:

(b_(#1,1)%3, b_(#1,2)%3),

(b_(#2,1)%3, b_(#2,2)%3),

(b_(#3,1)%3, b_(#3,2)%3), . . . ,

(b_(#k,1)%3, b_(#k,2)%3), . . . ,

(b_(#3g-2,1)%3, b_(#3g-2,2)%3),

(b_(#3g-1,1)%3, b_(#3g-1,2)%3), and

(b_(#3g,1)%3, b_(#3g,2)%3) are any of (1, 2), or (2, 1) (where k=1, 2,3, . . . , 3g).

<Condition #5-1> has a similar relationship with respect to equations19-1 to 19-3g as <Condition #2-1> has with respect to equations 15-1 to15-3g. If the condition below (<Condition #6-1>) is added for equations19-1 to 19-3g in addition to <Condition #5-1>, the possibility of beingable to create an LDPC-CC having higher error correction capability isincreased.

<Condition #6-1>

Orders of X(D) of equations 19-1 to 19-3g satisfy the followingcondition: all values other than multiples of 3 (that is, 0, 3, 6, . . ., 3g−3) from among integers from 0 to 3g−1 (0, 1, 2, 3, 4, . . . , 3g−2,3g−1) are present in the following 6g values of

(a_(#1,1,1)%3g, a_(#1,1,2)%3g),

(a_(#2,1,1)%3g, a_(#2,1,2)%3g), . . . ,

(a_(#p,1,1)%3g, a_(#p,1,2)%3g), . . . , and

(a_(#3g,1,1)%3g, a_(#3g,1,2)%3g) (where p=1, 2, 3, . . . , 3g); and

Orders of P(D) of equations 19-1 to 19-3g satisfy the followingcondition: all values other than multiples of 3 (that is, 0, 3, 6, . . ., 3g−3) from among integers from 0 to 3g−1 (0, 1, 2, 3, 4, . . . , 3g−2,3g−1) are present in the following 6g values of

(b_(#1,1)%3g, b_(#1,2)%3g),

(b_(#2,1)%3g, b_(#2,2)%3g),

(b_(#3,1)%3g, b_(#3,2)%3g), . . . ,

(b_(#k,1)%3g, b_(#k,2)%3g), . . . ,

(b_(#3g-2,1)%3g, b_(#3g-2,2)%3g),

(b_(#3g-1,1)%3g, b_(#3g-1,2)%3g), and

(b_(#3g,1)%3g, b_(#3g,2)%3g) (where k=1, 2, 3, . . . 3g).

The possibility of obtaining good error correction capability is high ifthere is also randomness while regularity is maintained for positions atwhich “1”s are present in a parity check matrix. With an LDPC-CC forwhich the time varying period is 3g (where g=2, 3, 4, 5, . . . ) and thecoding rate is ½ that has parity check polynomials of equations 19-1 to19-3g, if a code is created in which <Condition #6-1> is applied inaddition to <Condition #5-1>, it is possible to provide randomness whilemaintaining regularity for positions at which “1”s are present in aparity check matrix, so that the possibility of obtaining better errorcorrection capability is increased.

The possibility of being able to create a code having higher errorcorrection capability is also increased if a code is created using<Condition #6′-1> instead of <Condition #6-1>, that is, using <Condition#6′-1 in addition to <Condition #5-1>.

<Condition #6′-1>

Orders of X(D) of equations 19-1 to 19-3g satisfy the followingcondition: all values other than multiples of 3 (that is, 0, 3, 6, . . ., 3g−3) from among integers from 0 to 3g−1 (0, 1, 2, 3, 4, 3g−2, 3g−1)are present in the following 6g values of

(a_(#1,1,1)%3g, a_(#1,1,2)%3g),

(a_(#2,1,1)%3g, a_(#2,1,2)%3g), . . . ,

(a_(#p,1,1)%3g, a_(#p,1,2)%3g), . . . ,

and (a_(#3g,1,1)%3g, a_(#3g,1,2)%3g) (where p=1, 2, 3, . . . , 3g); or

Orders of P(D) of equations 19.1 to 19-3g satisfy the followingcondition: all values other than multiples of 3 (that is, 0, 3, 6, . . ., 3g−3) from among integers from 0 to 3g−1 (0, 1, 2, 3, 4, . . . , 3g−2,3g−1) are present in the following 6g values of

(b_(#1,1)%3g, b_(#1,2)%3g),

(b_(#2,1)%3g, b_(#2,2)%3g),

(b_(#3,1)%3g, b_(#3,2)%3g), . . . ,

(b_(#k,1)%3g, b_(#k,2)%3g), . . . ,

(b_(#3g-2,1)%3g, b_(#3g-2,2)%3g),

(b_(#3g-1,1)%3g, b_(#3g-1,2)%3g), and

(b_(#3g,1)%3g, b_(#3g,2)%3g) (where k=1, 2, 3, . . . , 3g).

Examples of LDPC-CCs of a coding rate of ½ and a time varying period of6 having good error correction capability are shown in Table 4.

TABLE 4 Code Parity check polynomial LDPC-CC #1 Check polynomial #1:(D³²⁸ + D³¹⁷ + 1)X(D) + (D⁵⁸⁹ + D⁴³⁴ + 1)P(D) = 0 of a time Checkpolynomial #2: (D⁵⁹⁶ + D⁵⁵³ + 1)X(D) + (D⁵⁸⁶ + D⁴⁶¹ + 1)P(D) = 0 varyingCheck polynomial #3: (D⁵⁵⁰ + D¹⁴³ + 1)X(D) + (D⁴⁷⁰ + D⁴⁴⁸ + 1)P(D) = 0period Check polynomial #4: (D⁴⁷⁰ + D²²³ + 1)X(D) + (D²⁵⁶ + D⁴¹ + 1)P(D)= 0 of 6 and Check polynomial #5: (D⁸⁹ + D⁴⁰ + 1)X(D) + (D³¹⁶ + D⁷¹ +1)P(D) = 0 a coding rate Check polynomial #6: (D³²⁰ + D¹⁹⁰ + 1)X(D) +(D⁵⁷⁵ + D¹³⁶ + 1)P(D) = 0 of ½ LDPC-CC #2 Check polynomial #1: (D⁵²⁴ +D⁵¹¹ + 1 )X(D) + (D²¹⁵ + D¹⁰³ + 1)P(D) = 0 of a time Check polynomial#2: (D⁵⁴⁷ + D²⁸⁷ + 1)X(D) + (D⁴⁶⁷ + D¹ + 1)P(D) = 0 varying Checkpolynomial #3: (D²⁸⁹ + D⁶² + 1)X(D) + (D⁵⁰³ + D⁵⁰² + 1)P(D) = 0 periodCheck polynomial #4: (D⁴⁰¹ + D⁵⁵ + 1)X(D) + (D⁴⁴³ + D¹⁰⁶ + 1)P(D) = 0 of6 and Check polynomial #5: (D⁴³³ + D³⁹⁵ + 1)X(D) + (D⁴⁰⁴ + D¹⁰⁰ + 1)P(D)= 0 a coding rate Check polynomial #6: (D¹³⁶ + D⁵⁹ + 1)X(D) + (D⁵⁹⁹ +D⁵⁵⁹ + 1)P(D) = 0 of ½ LDPC-CC #3 Check polynomial #1: (D²⁵³ + D⁴⁴ +1)X(D) + (D⁴⁷³ + D²⁵⁶ + 1)P(D) = 0 of a time Check polynomial #2:(D⁵⁹⁵ + D¹⁴³ + 1)X(D) + (D⁵⁹⁸ + D⁹⁵ + 1)P(D) = 0 varying Checkpolynomial #3: (D⁹⁷ + D¹¹ + 1)X(D) + (D⁵⁹² + D⁴⁹¹ + 1)P(D) = 0 periodCheck polynomial #4: (D⁵⁰ + D¹⁰ + 1)X(D) + (D³⁶⁸ + D¹¹² + 1)P(D) = 0 of6 and Check polynomial #5: (D²⁸⁶ + D²²¹ + 1)X(D) + (D⁵¹⁷ + D³⁵⁹ + 1)P(D)= 0 a coding rate Check polynomial #6: (D⁴⁰⁷ + D³²² + 1)X(D) + (D²⁸³ +D²⁵⁷ + 1)P(D) = 0 of ½

An LDPC-CC of a time varying period of g with good characteristics hasbeen described above. Also, for an LDPC-CC, it is possible to provideencoded data (codeword) by multiplying information vector n by generatormatrix G. That is, encoded data (codeword) c can be represented byc=n×G. Here, generator matrix G is found based on parity check matrix Hdesigned in advance. To be more specific, generator matrix G refers to amatrix satisfying G×H^(T)=0.

For example, a convolutional code of a coding rate of ½ and generatorpolynomial G=[1G₁(D)/G₀(D)] will be considered as an example. At thistime, G₁ represents a feed-forward polynomial and G₀ represents afeedback polynomial. If a polynomial representation of an informationsequence (data) is X(D), and a polynomial representation of a paritysequence is P(D), a parity check polynomial is represented as shown inequation 21 below.[21]G ₁(D)X(D)+G ₀(D)P(D)=0  (Equation 21)

where D is a delay operator.

FIG. 5 shows information relating to a (7, 5) convolutional code. A (7,5) convolutional code generator polynomial is represented as G=[1(D²+1)/(D²+D+1)]. Therefore, a parity check polynomial is as shown inequation 22 below.[22](D ²+1)X(D)+(D ² D+1)P(D)=0  (Equation 22)

Here, data at point in time i is represented by X_(i), and parity byP_(i), and transmission sequence Wi is represented as W_(i)=(X_(i),P_(i)). Then transmission vector w is represented as w=(X₁, P₁, X₂, P₂,. . . , X_(i), P_(i) . . . )^(T). Thus, from equation 22, parity checkmatrix H can be represented as shown in FIG. 5. At this time, therelational equation in equation 23 below holds true.[23]Hw=0  (Equation 23)

Therefore, using parity check matrix H, the decoding side can performdecoding using belief propagation such as BP (belief propagation)decoding, min-sum decoding which is approximation of BP decoding, offsetBP decoding, normalized BP decoding, shuffled BP decoding as shown inNon-Patent Literature 5 to Non-Patent Literature 7.

(Time-Invariant/Time Varying LDPC-CCs (of a Coding Rate of (n−1)/n)Based on a Convolutional Code (where n is a Natural Number))

An overview of time-invariant/time varying LDPC-CCs based on aconvolutional code is given below.

A parity check polynomial represented as shown in equation 24 will beconsidered, with polynomial representations of coding rate of R=(n−1)/nas information X₁, X₂, . . . , X_(n-1) as X₁(D), X₂(D), . . . ,X_(n-1)(D), and a polynomial representation of parity P as P(D).[24](D ^(a) ^(1,1) +D ^(a) ^(1,2) + . . . +D ^(a) ^(1/1) +1)X ₁(D)+(D ^(a)^(2,1) +D ^(a) ^(2,2) + . . . +D ^(a) ^(2/2) +1)X ₂(D)+ . . . +(D ^(a)^(n-1,1) +D ^(a) ^(n-1,2) + . . . +D ^(a) ^(n-1/n-1) 1)X _(n-1)(D)+(D^(b) ¹ +D ^(b) ² + . . . +D ^(b) ³ +1)P(D)=0  (Equation 24)

In equation 24, at this time, a_(p,p) (where p=1, 2, . . . , n−1 andq=1, 2, . . . , rp) is, for example, a natural number, and satisfies thecondition a_(p,1)≠a_(p,2)≠ . . . ≠a_(p,rp). Also, b_(q) (where q=1, 2, .. . , s) is a natural number, and satisfies the condition b₁≠b₂≠ . . .≠b_(g). A code defined by a parity check matrix based on a parity checkpolynomial of equation 24 at this time is called a time-invariantLDPC-CC here.

Here, m different parity check polynomials based on equation 24 areprovided (where m is an integer equal to or greater than 2). Theseparity check polynomials are represented as shown below.[25]A _(X1,i)(D)X ₁(D)+A _(X2,i)(D)X ₂(D)+ . . . +A _(Xn-1,i)(D)X_(n-1)(D)+B _(i)(D)P(D)=0  (Equation 25)

Here, i=0, 1, . . . , m−1.

Then information X₁, X₂, . . . , X_(n-1) at point in time j isrepresented as X_(1,j), X_(2,j), . . . , X_(n-1,j), parity P at point intime j is represented as P_(j), and u_(j)=(X_(1,j), X_(2,j), . . . ,X_(n-1,j), P_(j))^(T). At this time, information X_(1,j), X_(2,j), . . ., X_(n-1,j), and parity P_(j) at point in time j satisfy a parity checkpolynomial of equation 26.[26]A _(X1,k)(D)X ₁(D)+A _(X2,k)(D)X ₂(D)+ . . . +A _(Xn-1,k)(D)X_(n-1)(D)+B _(k)(D)P(D)=0 (k=j mod m)  (Equation 26)

Here, “j mod m” is a remainder after dividing j by m.

A code defined by a parity check matrix based on a parity checkpolynomial of equation 26 is called a time varying LDPC-CC here. At thistime, a time-invariant LDPC-CC defined by a parity check polynomial ofequation 24 and a time varying LDPC-CC defined by a parity checkpolynomial of equation 26 have a characteristic of enabling parityeasily to be found sequentially by means of a register and exclusive OR.

For example, the configuration of parity check matrix H of an LDPC-CC ofa time varying period of 2 and a coding rate of ⅔ based on equation 24to equation 26 is shown in FIG. 6. Two different check polynomials of atime varying period of 2 based on equation 26 are designed “checkequation #1” and “check equation #2.” In FIG. 6, (Ha, 111) is a partcorresponding to “check equation #1,” and (Hc, 111) is a partcorresponding to “check equation #2.” Below, (Ha, 111) and (Hc, 111) aredefined as sub-matrices.

Thus, LDPC-CC parity check matrix H of a time varying period of 2 ofthis proposal can be defined by a first sub-matrix representing a paritycheck polynomial of “check equation #1”, and by a second sub-matrixrepresenting a parity check polynomial of “check equation #2”.Specifically, in parity check matrix H, a first sub-matrix and secondsub-matrix are arranged alternately in the row direction. When thecoding rate is ⅔, a configuration is employed in which a sub-matrix isshifted three columns to the right between an i'th row and (i+1)-th row,as shown in FIG. 6.

In the case of a time varying LDPC-CC of a time varying period of 2, ani'th row sub-matrix and an (i+1)-th row sub-matrix are differentsub-matrices. That is to say, either sub-matrix (Ha, 111) or sub-matrix(Hc, 111) is a first sub-matrix, and the other is a second sub-matrix.If transmission vector u is represented as u=(X_(1,0), X_(2,0), P₀,X_(1,1), X_(2,1), P₁, . . . X_(1,k), X_(2,k), P_(k), . . . )^(T), therelationship Hu=0 holds true (see equation 23).

Next, an LDPC-CC for which the time varying period is m is considered inthe case of a coding rate of ⅔. In the same way as when the time varyingperiod is 2, m parity check polynomials represented by equation 24 areprovided. Then “check equation #1” represented by equation 24 isprovided. “Check equation #2” to “check equation #m” represented byequation 24 are provided in a similar way. Data X and parity P of pointin time mi+1 are represented by X_(mi+1) and P_(mi+1) respectively, dataX and parity P of point in time mi+2 are represented by X_(mi+2) andP_(mi+2) respectively, . . . , and data X and parity P of point in timemi+m are represented by X_(mi+m) and P_(mi+m) respectively (where i isan integer).

Consider an LDPC-CC for which parity P_(mi+1) of point in time mi+1 isfound using “check equation #1,” parity P_(mi+2) of point in time mi+2is found using “check equation #2,” . . . , and parity P_(mi+m) of pointin time mi+m is found using “check equation #m.” An LDPC-CC code of thiskind provides the following advantages:

An encoder can be configured easily, and parity can be foundsequentially.

Termination bit reduction and received quality improvement in puncturingupon termination can be expected.

FIG. 7 shows the configuration of the above LDPC-CC parity check matrixof a coding rate of ⅔ and a time varying period of m. In FIG. 7, (H₁,111) is a part corresponding to “check equation #1,” (H₂, 111) is a partcorresponding to “check equation #2,” . . . , and (H_(m), 111) is a partcorresponding to “check equation #m.” Below, (H₁, 111) is defined as afirst sub-matrix, (H₂, 111) is defined as a second sub-matrix, . . . ,and (H_(m), 111) is defined as an m-th sub-matrix.

Thus, LDPC-CC parity check matrix H of a time varying period of m ofthis proposal can be defined by a first sub-matrix representing a paritycheck polynomial of “check equation #1”, a second sub-matrixrepresenting a parity check polynomial of “check equation #2”, . . . ,and an m-th sub-matrix representing a parity check polynomial of “checkequation #m”. Specifically, in parity check matrix H, a first sub-matrixto m-th sub-matrix are arranged periodically in the row direction (seeFIG. 7). When the coding rate is ⅔, a configuration is employed in whicha sub-matrix is shifted three columns to the right between an i-th rowand (i+1)-th row (see FIG. 7).

If transmission vector u is represented as u=(X_(1,0), X_(2,0), P₀,X_(1,1), X_(2,1), P₁, . . . , X_(1,k), X_(2,k), P_(k), . . . )^(T), therelationship Hu=0 holds true (see equation 23).

In the above description, a case of a coding rate of ⅔ has beendescribed as an example of a time-invariant/time varying LDPC-CC basedon a convolutional code of a coding rate of (n−1)/n, but atime-invariant/time varying LDPC-CC parity check matrix based on aconvolutional code of a coding rate of (n−1)/n can be created bythinking in a similar way.

That is to say, in the case of a coding rate of ⅔, in FIG. 7, (H₁, 111)is a part (first sub-matrix) corresponding to “check equation #1,” (H₂,111) is a part (second sub-matrix) corresponding to “check equation #2,”. . . , and (H_(m), 111) is a part (m-th sub-matrix) corresponding to“check equation #m,” while, in the case of a coding rate of (n−1)/n, thesituation is as shown in FIG. 8. That is to say, a part (firstsub-matrix) corresponding to “check equation #1” is represented by (H₁,11 . . . 1), and a part (k-th sub-matrix) corresponding to “checkequation #k” (where k=2, 3, . . . , m) is represented by (H_(k), 11 . .. 1). At this time, the number of “1”s of parts excluding H_(k) in thek-th sub-matrix is n. Also, in parity check matrix H, a configuration isemployed in which a sub-matrix is shifted n columns to the right betweenan i'th row and (i+1)-th row (see FIG. 8).

If transmission vector u is represented as u=(X_(1,0), X_(2,0), . . . ,X_(n-1,0), P₀, X_(1,1), X_(2,1), . . . , X_(n-1,1), P₁, . . . , X_(1,k),X_(2,k), . . . , X_(n-1,k), P_(k), . . . )^(T), the relationship Hu-0holds true (see equation 23).

FIG. 9 shows an example of the configuration of an LDPC-CC encoder whenthe coding rate is R=½. As shown in FIG. 9, LDPC-CC encoder 100 isprovided mainly with data computing section 110, parity computingsection 120, weight control section 130, and modulo 2 adder (exclusiveOR computer) 140.

Data computing section 110 is provided with shift registers 111-1 to111-M and weight multipliers 112-0 to 112-M.

Parity computing section 120 is provided with shift registers 121-1 to121-M and weight multipliers 122-0 to 122-M.

Shift registers 111-1 to 111-M and 121-1 to 121-M are registers storingv_(1,t-i) and v_(2,t-i) (where i=0, . . . , M) respectively, and, at atiming at which the next input comes in, send a stored value to theadjacent shift register to the right, and store a new value sent fromthe adjacent shift register to the left. The initial state of the shiftregisters is all-zeros.

Weight multipliers 112-0 to 112-M and 122-0 to 122-M switch values of h₁^((m)) and h₂ ^((m)) to 0 or 1 in accordance with a control signaloutputted from weight control section 130.

Based on a parity check matrix stored internally, weight control section130 outputs values of h₁ ^((m)) and h₂ ^((m)) at that timing, andsupplies them to weight multipliers 112-0 to 112-M and 122-0 to 122-M.

Modulo 2 adder 140 adds all modulo 2 calculation results to the outputsof weight multipliers 112-0 to 112-M and 122-0 to 122-M, and calculatesp_(i).

By employing this kind of configuration, LDPC-CC encoder 100 can performLDPC-CC encoding in accordance with a parity check matrix.

If the arrangement of rows of a parity check matrix stored by weightcontrol section 130 differs on a row-by-row basis, LDPC-CC encoder 100is a time varying convolutional encoder. Also in the case of an LDPC-CCof a coding rate of (q−1)/q, a configuration needs to be employed inwhich (q−1) data computing sections 110 are provided and modulo 2 adder140 performs modulo 2 addition (exclusive OR computation) of the outputsof weight multipliers.

Embodiment 2

Next, the present embodiment will explain a search method that cansupport a plurality of coding rates in a low computational complexity inan encoder and decoder. By using an LDPC-CC searched out by the methoddescribed below, it is possible to realize high data received quality inthe decoder.

With the LDPC-CC search method according to the present embodiment,LDPC-CCs of coding rates of ⅔, ¾, ⅘, . . . , (q−1)/q are sequentiallysearched based on, for example, a coding rate of ½ among LDPC-CCs withgood characteristics described above. By this means, in coding anddecoding processing, by preparing a coder and decoder in the highestcoding rate of (q−1)/q, it is possible to perform coding and decoding ina coding rate of (s−1)/s (S=2, 3, . . . , q−1) lower than the highestcoding rate of (q−1)/q.

A case in which the time varying period is 3 will be described below asan example. As described above, an LDPC-CC for which the time varyingperiod is 3 can provide excellent error correction capability.

(LDPC-CC Search Method)

(1) Coding Rate of ½

First, an LDPC-CC of a coding rate of ½ is selected as a referenceLDPC-CC of a coding rate of ½. Here, an LDPC-CC of good characteristicsdescribed above is selected as a reference LDPC-CC of a coding rate of½.

A case will be explained below where the parity check polynomialsrepresented by equations 27-1 to 27-3 are used as parity checkpolynomials of a reference LDPC-CC of a coding rate of ½. The examplesof equations 27-1 to 27-3 are represented in the same way as above (i.e.an LDPC-CC of good characteristics), so that it is possible to define anLDPC-CC of a time varying period of 3 by three parity check polynomials[27](D ³⁷³ +D ⁵⁶+1)X ₁(D)+(D ⁴⁰⁶ +D ²¹⁸+1)P(D)=0  (Equation 27-1)(D ⁴⁵⁷ +D ¹⁹⁷+1)X ₁(D)+(D ⁴⁹¹ +D ²²+1)P(D)=0  (Equation 27-2)(D ⁴⁸⁵ +D ⁷⁰+1)X ₁(D)+(D ²³⁶ +D ¹⁸¹+1)P(D)=0  (Equation 27-3)

As described in table 3, equations 27-1 to 27-3 are represented as anexample of an LDPC-CC with good characteristics where the time varyingperiod is 3 and the coding rate is ½. Then, as described above (with anLDPC-CC of good characteristics), information X₁ at point in time j isrepresented as X_(1,j), parity P at point in time j is represented asP_(j), and u_(j)=(X_(1,j), P_(j))^(T). At this time, information X_(1,j)and parity P_(j) at point in time j satisfy a parity check polynomial ofequation 27-1 when “j mod 3=0.” Further, information X_(1,j) and parityP_(j) at point in time j satisfy a parity check polynomial of equation27-2 when “j mod 3=1.” Further, information X_(1,j) and parity P_(j) atpoint in time j satisfy a parity chock polynomial of equation 27-3 when“j mod 3=2.” At this time, the relationships between panty checkpolynomials and a parity check matrix are the same as above (i.e. as inan LDPC-CC of good characteristics).

(2) Coding Rate of ⅔

Next, LDPC-CC parity check polynomials of a coding rate of ⅔ is createdbased on the parity check polynomials of a coding rate of ½ with goodcharacteristics. To be more specific, LDPC-CC parity check polynomialsof a coding rate of ⅔ are formed, including the reference parity checkpolynomials of a coding rate of ½.

As shown in equations 28-1 to 28-3, upon using equations 27-1 to 27-3 ina reference LDPC-CC of a coding rate of ½, it is possible to representLDPC-CC parity check polynomials of a coding rate of ⅔.[28](D ³⁷³ +D ⁵⁶+1)X ₁(D)+(D ^(α1) +D ^(β1)+1)X ₂(D)+(D ⁴⁰⁶ +D²¹⁸+1)P(D)=0  (Equation 28-1)(D ⁴⁵⁷ +D ¹⁹⁷+1)X ₁(D)+(D ^(α2) +D ^(β2)+1)X ₂(D)+(D ⁴⁹¹ +D²²²+1)P(D)=0  (Equation 28-2)(D ⁴⁸⁵ +D ⁷⁰+1)X ₁(D)+(D ^(α3) +D ^(β3)+1)X ₂(D)+(D ²³⁶ +D¹⁸¹+1)P(D)=0  (Equation 28-3)

The parity check polynomials represented by equations 28-1 to 28-3 areformed by adding term X₂(D) to equations 27-1 to 27-3. LDPC-CC paritycheck polynomials of a coding rate of ⅔ used in equations 28-1 to 28-3are references for parity check polynomials of a coding rate of ¾.

Also, in equations 28-1 to 28-3, if the orders of X₂(D), (α1,β1),(α2,β2), (α3, β3), are set to satisfy the above conditions (e.g.<Condition #1> to <Condition #6>), it is possible to provide an LDPC-CCof good characteristics even in a coding rate of ⅔.

Then, as described above (with an LDPC-CC of good characteristics),information X₁ and X₂ at point in time j is represented as X_(1,j) andX_(2,j), parity P at point in time j is represented as P_(j), andu_(j)=(X_(1,j), X_(2,j), P_(j))^(T). At this time, information X_(1,j)and X_(2,j) and parity P_(j) at point in time j satisfy a parity checkpolynomial of equation 28-1 when “j mod 3=0.” Further, informationX_(1,j) and X_(2,j) and parity P_(j) at point in time j satisfy a paritycheck polynomial of equation 28-2 when “j mod 3=1.” Further, informationX_(1,j) and X_(2,j) and parity P_(j) at point in time j satisfy a paritycheck polynomial of equation 28-3 when “j mod 3=2.” At this time, therelationships between parity check polynomials and a parity check matrixare the same as above (i.e. as in an LDPC-CC of good characteristics).

(3) Coding Rate of ¾

Next, LDPC-CC parity check polynomials of a coding rate of ¾ is createdbased on the above parity check polynomials of a coding rate of ⅔. To bemore specific, LDPC-CC parity check polynomials of a coding rate of ¾are formed, including the reference parity check polynomials of a codingrate of ⅔.

Equations 29-1 to 29-3 show LDPC-CC parity check polynomials of a codingrate of ¾ upon using equations 28-1 to 28-3 in a reference LDPC-CC of acoding rate of ⅔.[29](D ³⁷³ +D ⁵⁶+1)X ₁(D)+(D ^(α1) +D ^(β1)+1)X ₂(D)+(D ^(γ1) +D ^(δ1)+1)X₃(D)+(D ⁴⁰⁶ +D ²¹⁸+1)P(D)=0  (Equation 29-1)(D ⁴⁵⁷ +D ¹⁹⁷+1)X ₁(D)+(D ^(α2) +D ^(β2)+1)X ₂(D)+(D ^(γ2) +D ^(δ2)+1)X₃(D)+(D ⁴⁹¹ +D ²²+1)P(D)=0  (Equation 29-2)(D ⁴⁸⁵ +D ⁷⁰+1)X ₁(D)+(D ^(α3) +D ^(β3)+1)X ₂(D)+(D ^(γ3) +D ^(δ3)+1)X₃(D)+(D ²⁵⁶ +D ¹⁸¹+1)P(D)=0  (Equation 29-3)

The parity check polynomials represented by equations 29-1 to 29-3 areformed by adding term X₃(D) to equations 28-1 to 28-3. Also, inequations 29-1 to 29-3, if the orders in X₃(D), (γ1, δ1), (γ2, δ2), (γ3,δ3), are set to satisfy the above conditions (e.g. <Condition #1> to<Condition #6>) with good characteristics, it is possible to provide anLDPC-CC of good characteristics even in a coding rate of ¾.

Then, as described above (LDPC-CC of good characteristics), informationX₁, X₂ and X₃ at point in time j is represented as X_(1,j), X_(2,j) andX_(3,j), parity P at point in time j is represented as P_(j), andu_(j)=(X_(1,j), X_(2,j), X_(3,j), P_(j))^(T). At this time, informationX_(1,j), X_(2,j) and X_(3,j) and parity P_(j) at point in time j satisfya parity check polynomial of equation 29-1 when “j mod 3=0.” Further,information X_(1,j), X_(2,j) and X_(3,j) and parity P_(j) at point intime j satisfy a parity check polynomial of equation 29-2 when “j mod3=1.” Further, information X_(1,j), X_(2,j) and X_(3,j) and parity P_(j)at point in time j satisfy a parity check polynomial of equation 29-3when “j mod 3=2.” At this time, the relationships between parity checkpolynomials and a parity check matrix are the same as above (i.e. as inan LDPC-CC of good characteristics).

Equations 30-1 to 30-(q−1) show general LDPC-CC parity check polynomialsof a time varying period of g upon performing a search as above.

$\begin{matrix}{\mspace{20mu}\lbrack 30\rbrack} & \; \\{{{{A_{{X\; 1},k}(D)}{X_{1}(D)}} + {{B_{k}(D)}{P(D)}}} = {0\mspace{14mu}( {k = {i\;{mod}\; g}} )}} & ( {{Equation}\mspace{14mu} 30\text{-}1} ) \\{{{{{{A_{{X\; 1},k}(D)}{X_{1}(D)}} + {{A_{{X\; 2},k}(D)}{X_{2}(D)}} + {{B_{k}(D)}{P(D)}}} = 0}\;\quad}\mspace{11mu}( {k = {i\;{mod}\; g}} )} & ( {{Equation}\mspace{14mu} 30\text{-}2} ) \\{{{{A_{{X\; 1},k}(D)}{X_{1}(D)}} + {{A_{{X\; 2},k}(D)}{X_{2}(D)}} + {{A_{{X\; 3},k}(D)}{X_{3}(D)}} + {{B_{k}(D)}{P(D)}}} = {0\mspace{14mu}( {k = {i\;{mod}\; g}} )}} & ( {{Equation}\mspace{14mu} 30\text{-}3} ) \\{\mspace{20mu}\vdots} & \; \\{{{{A_{{X\; 1},k}(D)}{X_{1}(D)}} + {{A_{{X\; 2},k}(D)}{X_{2}(D)}} + \ldots + {{A_{{{X\; q} - 1},k}(D)}{X_{q - 1}(D)}} + {{B_{k}(D)}{P(D)}}} = {0\mspace{14mu}( {k = {i\;{mod}\; g}} )}} & ( {{Equation}\mspace{14mu} 30\text{-}( {q - 1} )} )\end{matrix}$

Here, equation 30-1 is represented as above because it is a generalequation. However, as described above (with an LDPC-CC of goodcharacteristics), the time varying period is g, and therefore equation30-1 is actually represented by g parity check polynomials. For example,as described with the present embodiment, when the time varying periodis 3, representation of three parity check polynomials is provided asshown in equations 27-1 to 27-3. Similar to equation 30-1, equations30-2 to 30-(q−1) each have a time varying period of g, and therefore arerepresented by g parity check equations.

Here, assume that g parity check equations of equation 30-1 arerepresented by equation 30-1-0, equation 30-1-1, equation 30-1-2, . . ., equation 30-1-(g−2) and equation 30-1-(g−1).

Similarly, equation 30-w is represented by g parity check polynomials(w=2, 3, . . . , q−1). Here, assume that g parity check equations ofequation 30-w are represented by equation 30-w-0, equation 30-w-1,equation 30-w-2, . . . , equation 30-w-(g−2) and equation 30-w-(g−1).

Also, in equations 30-1 to 30-(q−1), information X₁, X₂, . . . , X_(q-1)at point in time i is represented as X_(1,i), X_(2,i), . . . ,X_(q-1,i), and parity P at point in time i is represented as P_(i).Also, A_(Xr,k)(D) refers to a term of X_(r)(D) in the parity checkpolynomial for k calculated from “k=i mod g,” at point in time i wherethe coding rate is (r−1)/r (r=2, 3, . . . , q, and q is a natural numberequal to or greater than 3). Also, B_(k)(D) refers to a term of P(D) inthe parity check polynomial for k calculated from “k=i mod g,” at pointin time i where the coding rate is (r−1)/r. Here, “i mod g” is aremainder after dividing i by g.

That is, equation 30-1 represents an LDPC-CC parity check polynomial ofa time varying period of g supporting a coding rate of ½, equation 30-2represents an LDPC-CC parity check polynomial of a time varying periodof g supporting a coding rate of ⅔, . . . , and equation 30-(q−1)represents an LDPC-CC parity check polynomial of a time varying periodof g supporting a coding rate of (q−1)/q.

Thus, based on equation 30-1 which represents an LDPC-CC parity checkpolynomial of a coding rate of ½ with good characteristics, an LDPC-CCparity check polynomial of a coding rate of ⅔ (i.e. equation 30-2) isgenerated.

Further, based on equation 30-2 which represents an LDPC-CC parity checkpolynomial of a coding rate of ⅔, an LDPC-CC parity check polynomial ofa coding rate of ¾ (i.e. equation 30-3) is generated. The same appliesto the following, and, based on an LDPC-CC of a coding rate of (r−1)/r,LDPC-CC parity check polynomials of a coding rate of r/(r+1) (r=2, 3, .. . , q−2, q−1) are generated.

The above method of forming parity check polynomials will be shown in adifferent way. Consider an LDPC-CC for which the coding rate is (y−1)/yand the time varying period is g, and an LDPC-CC for which the codingrate is (z−1)/z and the time varying period is g. Here, the maximumcoding rate is (q−1)/q among coding rates to share encoder circuits andto share decoder circuits, where g is an integer equal to or greaterthan 2, y is an integer equal to or greater than 2, z is an integerequal to or greater than 2, and the relationship of y<z≦q holds true.Here, sharing encoder circuits means to share circuits inside encoders,and does not mean to share circuits between an encoder and a decoder.

At this time, if w=y−1 is assumed in equations 30-w-0, 30-w-1, 30-w-2, .. . , 30-w-(g−2) and 30-w-(g−1), which represent g parity checkpolynomials described upon explaining equations 30-1 to 30-(q−1),representations of g parity check polynomials is provided as shown inequations 31-1 to 31-g.

$\begin{matrix}{\mspace{20mu}\lbrack 31\rbrack} & \; \\{{{{A_{{X\; 1},0}(D)}{X_{1}(D)}} + {{A_{{X\; 2},0}(D)}{X_{2}(D)}} + \ldots + {{A_{{{X\; y} - 1},0}(D)}{X_{y - 1}(D)}} + {{B_{0}(D)}{P(D)}}} = {0\mspace{14mu}( {0 = {i\;{mod}\; g}} )}} & ( {{Equation}\mspace{14mu} 31\text{-}1} ) \\{{{{B_{0}(D)}{P(D)}} + {\sum\limits_{r = 1}^{y - 1}{{A_{{X\; r},0}(D)}{X_{r}(D)}}}} = {0\mspace{14mu}( {0 = {i\;{mod}\; g}} )}} & ( {{Equation}\mspace{14mu} 31\text{-}1^{\prime}} ) \\{{{{A_{{X\; 1},1}(D)}{X_{1}(D)}} + {{A_{{X\; 2},1}(D)}{X_{2}(D)}} + \ldots + {{A_{{{X\; y} - 1},1}(D)}{X_{y - 1}(D)}} + {{B_{1}(D)}{P(D)}}} = {0\mspace{14mu}( {1 = {i\;{mod}\; g}} )}} & ( {{Equation}\mspace{14mu} 31\text{-}2} ) \\{{{{B_{1}(D)}{P(D)}} + {\sum\limits_{r = 1}^{y - 1}{{A_{{X\; r},1}(D)}{X_{r}(D)}}}} = {0\mspace{14mu}( {1 = {i\;{mod}\; g}} )}} & ( {{Equation}\mspace{14mu} 31\text{-}2^{\prime}} ) \\{\mspace{20mu}\vdots} & \; \\{{{{A_{{X\; 1},k}(D)}{X_{1}(D)}} + {{A_{{X\; 2},k}(D)}{X_{2}(D)}} + \ldots + {{A_{{{X\; y} - 1},k}(D)}{X_{y - 1}(D)}} + {{B_{k}(D)}{P(D)}}} = {0\mspace{14mu}( {k = {i\;{mod}\; g}} )}} & ( {{Equation}\mspace{14mu} 31\text{-}( {k + 1} )} ) \\{{{{B_{k}(D)}{P(D)}} + {\sum\limits_{r = 1}^{y - 1}{{A_{{X\; r},k}(D)}{X_{r}(D)}}}} = {0\mspace{14mu}( {k = {i\;{mod}\; g}} )}} & ( {{Equation}\mspace{14mu} 31\text{-}( {k + 1} )^{\prime}} ) \\{\mspace{20mu}\vdots} & \; \\{{{{A_{{X\; 1},{g - 1}}(D)}{X_{1}(D)}} + {{A_{{X\; 2},{g - 1}}(D)}{X_{2}(D)}} + \ldots + {{A_{{{X\; y} - 1},{g - 1}}(D)}{X_{y - 1}(D)}} + {{B_{g - 1}(D)}{P(D)}}} = {0\mspace{14mu}( {{g - 1} = {i\;{mod}\; g}} )}} & ( {{Equation}\mspace{14mu} 31\text{-}g} ) \\{{{{{{B_{g - 1}(D)}{P(D)}} + {\sum\limits_{r = 1}^{y - 1}{{A_{{X\; r},{g - 1}}(D)}{X_{r}(D)}}}} = 0}\quad}\mspace{14mu}( {{g - 1} = {i\;{mod}\; g}} )} & ( {{Equation}\mspace{14mu} 31\text{-}g^{\prime}} )\end{matrix}$

In equations 31-1 to 31-g, equation 31-w and equation 31-w′ areequivalent, and therefore it is possible to replace equation 31-w belowwith equation 31-w′ (w=1, 2, . . . , g).

Then, as described above (with an LDPC-CC of good characteristics),information X₁, X₂, . . . , X_(y-1) at point in time j is represented asX_(1,j), X_(2,j), . . . , X_(y-1,j), parity P at point in time j isrepresented as P_(j), and u_(j)=(X_(1,j), X_(2,j), . . . , X_(y-1,j),P_(j))^(T). At this time, information X_(1,j), X_(2,j), . . . , X_(y-1)and parity P_(j) at point in time j: satisfy a parity check polynomialof equation 31-1 when “j mod g=0”; satisfy a parity check polynomial ofequation 31-2 when “j mod g=1”; satisfy a parity check polynomial ofequation 31-3 when “j mod g=2”; . . . ; satisfy a parity checkpolynomial of equation 31-(k+1) when “j mod g=k”; . . . ; and satisfy aparity check polynomial of equation 31-g when “5 mod g=g−1.” At thistime, the relationships between parity check polynomials and a paritycheck matrix are the same as above (i.e. as in an LDPC-CC of goodcharacteristics).

Next, if w=z−1 is assumed in equations 30-w-0, 30-w-1, 30-w-2, . . . ,30-w-(g−2) and 30-w-(g−1), which represent g parity check polynomialsdescribed upon explaining equations 30-1 to 30-(q−1), representations ofg parity check polynomials can be provided as shown in equations 32-1 to32-g. Here, from the relationship of y<z≦q, representations of equations32-1 to 32-g can be provided.

$\begin{matrix}{\mspace{20mu}\lbrack 32\rbrack} & \; \\{{{{A_{{X\; 1},0}(D)}{X_{1}(D)}} + {{A_{{X\; 2},0}(D)}{X_{2}(D)}} + \ldots + {{A_{{{X\; y} - 1},0}(D)}{X_{y - 1}(D)}} + \ldots + {{A_{{X\; s},0}(D)}{X_{s}(D)}} + \ldots + {{A_{{{X\; z} - 1},0}(D)}{X_{z - 1}(D)}} + {{B_{0}(D)}{P(D)}}} = {0\mspace{14mu}( {0 = {i\;{mod}\; g}} )}} & ( {{Equation}\mspace{14mu} 32\text{-}1} ) \\{{{B_{0}(D)}{P(D)}} + {\sum\limits_{r = 1}^{y - 1}{{A_{{X\; r},0}(D)}{X_{r}(D)}}} + {\quad{{\sum\limits_{s = y}^{z - 1}{{A_{{X\; s},0}(D)}{X_{s}(D)}}} = {0\mspace{14mu}( {0 = {i\;{mod}\; g}} )}}}} & ( {{Equation}\mspace{14mu} 32\text{-}1^{\prime}} ) \\{{{{A_{{X\; 1},1}(D)}{X_{1}(D)}} + {{A_{{X\; 2},1}(D)}{X_{2}(D)}} + \ldots + {{A_{{{X\; y} - 1},1}(D)}{X_{y - 1}(D)}} + \ldots + {{A_{{X\; s},1}(D)}{X_{s}(D)}} + \ldots + {{A_{{{X\; z} - 1},1}(D)}{X_{z - 1}(D)}} + {{B_{1}(D)}{P(D)}}} = {0\mspace{14mu}( {1 = {i\;{mod}\; g}} )}} & ( {{Equation}\mspace{14mu} 32\text{-}2} ) \\{{{B_{1}(D)}{P(D)}} + {\sum\limits_{r = 1}^{y - 1}{{A_{{X\; r},1}(D)}{X_{r}(D)}}} + {\quad{{\sum\limits_{s = y}^{z - 1}{{A_{{X\; s},1}(D)}{X_{s}(D)}}} = {0\mspace{14mu}( {1 = {i\;{mod}\; g}} )}}}} & ( {{Equation}\mspace{14mu} 32\text{-}2^{\prime}} ) \\{\mspace{20mu}\vdots} & \; \\{{{{A_{{X\; 1},k}(D)}{X_{1}(D)}} + {{A_{{X\; 2},k}(D)}{X_{2}(D)}} + \ldots + {{A_{{{X\; y} - 1},k}(D)}{X_{y - 1}(D)}} + \ldots + {{A_{{X\; s},k}(D)}{X_{s}(D)}} + \ldots + {{A_{{{X\; z} - 1},k}(D)}{X_{z - 1}(D)}} + {{B_{k}(D)}{P(D)}}} = {0\mspace{14mu}( {k = {i\;{mod}\; g}} )}} & ( {{Equation}\mspace{14mu} 32\text{-}( {k + 1} )} ) \\{{{B_{k}(D)}{P(D)}} + {\sum\limits_{r = 1}^{y - 1}{{A_{{X\; r},k}(D)}{X_{r}(D)}}} + {\quad{{\sum\limits_{s = y}^{z - 1}{{A_{{X\; s},k}(D)}{X_{s}(D)}}} = {0\mspace{14mu}( {k = {i\;{mod}\; g}} )}}}} & ( {{Equation}\mspace{14mu} 32\text{-}( {k + 1} )^{\prime}} ) \\{\mspace{20mu}\vdots} & \; \\{{{{A_{{X\; 1},{g - 1}}(D)}{X_{1}(D)}} + {{A_{{X\; 2},{g - 1}}(D)}{X_{2}(D)}} + \ldots + {{A_{{{X\; y} - 1},{g - 1}}(D)}{X_{y - 1}(D)}} + \ldots + {{A_{{X\; s},{g - 1}}(D)}{X_{s}(D)}} + \ldots + {{A_{{{X\; z} - 1},{g - 1}}(D)}{X_{z - 1}(D)}} + {{B_{g - 1}(D)}{P(D)}}} = {0\mspace{14mu}( {{g - 1} = {i\;{mod}\; g}} )}} & ( {{Equation}\mspace{14mu} 32\text{-}g} ) \\{{{B_{g - 1}(D)}{P(D)}} + {\sum\limits_{r = 1}^{y - 1}{{A_{{X\; r},{g - 1}}(D)}{X_{r}(D)}}} + {\quad{{\sum\limits_{s = y}^{z - 1}{{A_{{X\; s},{g - 1}}(D)}{X_{s}(D)}}} = {0\mspace{14mu}( {{g - 1} = {i\;{mod}\; g}} )}}}} & ( {{Equation}\mspace{14mu} 32\text{-}g^{\prime}} )\end{matrix}$

In equations 32-1 to 32-g, equation 32-w and equation 32-w′ areequivalent, and therefore it is possible to replace equation 32-w belowwith equation 32-w′ (w=1, 2, . . . , g).

Then, as described above (LDPC-CC of good characteristics), informationX₁, X₂, . . . , X_(y-1), . . . , X_(s), . . . , X_(z-1) at point in timej is represented as X_(1,j), X_(2,j), . . . , X_(y-1,j), . . . ,X_(s,j), . . . , X_(z-1,j), parity P at point in time j is representedas P_(j), and u_(j)=(X_(1,j), X_(2,j), . . . , X_(y-1,j), . . . ,X_(G,j), . . . , X_(z-1,j), P_(j))^(T) (here, from the relationship ofy<z≦q, s=y, y+1, y+2, y+3, . . . , z−3, z−2, z−1). At this time,information X_(1,j), X_(2,j), . . . , X_(y-1,j), . . . , X_(s,j), . . .X_(z-1,j) and parity P_(j) at point in time j: satisfy a parity checkpolynomial of equation 32-1 when “j mod g=0”; satisfy a parity checkpolynomial of equation 32-2 when “j mod g=1”; satisfy a parity checkpolynomial of equation 32-3 when “j mod g=2”; . . . , satisfy a paritycheck polynomial of equation 32-(k+1) when “j mod g=k”; . . . ; andsatisfy a parity check polynomial of equation 32-g when “j mod g=g−1.”At this time, the relationships between parity check polynomials and aparity check matrix are the same as above (i.e. as in an LDPC-CC of goodcharacteristics).

In a case where the above relationships hold true, if the followingconditions hold true for an LDPC-CC of a time varying period of g in acoding rate of (y−1)/y and for an LDPC-CC of a time varying period of gin a coding rate of (z−1)/z, it is possible to share circuits between anencoder for an LDPC-CC of a time varying period of g in a coding rate of(y−1)/y and an encoder for an LDPC-CC of a time varying period of g in acoding rate of (z−1)/z, and it is possible to share circuits between adecoder for an LDPC-CC of a time varying period of g in a coding rate of(y−1)/y and a decoder for an LDPC-CC of a time varying period of g in acoding rate of (z−1)/z. The conditions are as follows.

First, the following relationships hold true between equation 31-1 andequation 32-1:

A_(X1,0)(D) of equation 31-1 and A_(X1,0)(D) of equation 32-1 are equal;

A_(Xf,0)(D) of equation 31-1 and A_(Xf,0)(D) of equation 32-1 are equal;

-   -   . . .

A_(Xy-1,0)(D) of equation 31-1 and A_(Xy-1,0)(D) of equation 32-1 areequal. That is, the above relationships hold true for f−1, 2, 3, . . . ,y−1.

Also, the following relationship holds true for parity:

B₀(D) of equation 31-1 and B₀(D) of equation 32-1 are equal.

Similarly, the following relationships hold true between equation 31-2and equation 32-2:

A_(X1,1)(D) of equation 31-2 and A_(X1,1)(D) of equation 32-2 are equal;

-   -   . . .

A_(Xf,1)(D) of equation 31-2 and A_(Xf,1)(D) of equation 32-2 are equal;

-   -   . . .

A_(Xy-1,1)(D) of equation 31-2 and A_(Xy-1,1)(D) of equation 32-2 areequal. That is, the above relationships hold true for f=1, 2, 3, . . . ,y−1.

Also, the following relationship holds true for parity:

B₁(D) of equation 31-2 and B₁(D) of equation 32-2 are equal, and so on.

Similarly, the following relationships hold true between equation 31-hand equation 32-h:

A_(X1,h-1)(D) of equation 31-h and A_(X1,h-1)(D) of equation 32-h areequal;

A_(Xf,h-1)(D) of equation 31-h and A_(Xf,h-1)(D) of equation 32-h areequal;

-   -   . . .

A_(Xy-1,h-1)(D) of equation 31-h and A_(Xy-1,h-1)(D) of equation 32-hare equal. That is, the above relationships hold true for f=1, 2, 3, . .. , y−1.

Also, the following relationship holds true for parity:

B₁(D) of equation 31-h and B₁(D) of equation 32-h are equal, and so on.

Similarly, the following relationships hold true between equation 31-gand equation 32-g:

A_(X1,g-1)(D) of equation 31-g and A_(X1,g-1)(D) of equation 32-g areequal;

-   -   . . .

A_(Xf,g-1)(D) of equation 31-g and A_(Xf,g-1)(D) of equation 32-g areequal;

-   -   . . .

A_(Xy-1,g-1)(D) of equation 31-g and A_(Xy-1,g-1)(D) of equation 32-gare equal. That is, the above relationships hold true for f=1, 2, 3, . .. , y−1.

Also, the following relationship holds true for parity:

B_(g-1)(D) of equation 31-g and B_(g-1)(D) of equation 32-g are equal(therefore, h=1, 2, 3, . . . , g−2, g−1, g).

In a case where the above relationships hold true, it is possible toshare circuits between an encoder for an LDPC-CC of a time varyingperiod of g in a coding rate of (y−1)/y and an encoder for an LDPC-CC ofa time varying period of g in a coding rate of (z−1)/z, and it ispossible to share circuits between a decoder for an LDPC-CC of a timevarying period of g in a coding rate of (y−1)/y and a decoder for anLDPC-CC of a time varying period of g in a coding rate of (z−1)/z. Here,the method of sharing encoder circuits and the method of sharing decodercircuits will be explained in detail in the following (configurations ofan encoder and decoder).

Examples of LDPC-CC parity check polynomials will be shown in table 5,where the time varying period is 3 and the coding rate is ½, ⅔, ¾ or ⅚.Here, the form of parity check polynomials is the same as in the form oftable 3. By this means, if the transmitting apparatus and the receivingapparatus support coding rates of ½, ⅔, ¾ and ⅚ (or if the transmittingapparatus and the receiving apparatus support two or more of the fourcoding rates), it is possible to reduce the computational complexity(circuit scale) (this is because it is possible to share encodercircuits and decoder circuits even in the case of distributed codes, andtherefore reduce the circuit scale), and provide data of high receivedquality in the receiving apparatus.

TABLE 5 Code Parity check polynomial LDPC-CC Check polynomial #1:(D³⁷³ + D⁵⁶ + 1)X₁(D) + (D⁴⁰⁶ + D²¹⁸ + 1)P(D) = 0 of a time Checkpolynomial #2: (D⁴⁵⁷ + D¹⁹⁷ + 1)X₁(D) + (D⁴⁹¹ + D²² + 1)P(D) = 0 varyingCheck polynomial #3: (D⁴⁸⁵ + D⁷⁰ + 1)X₁(D) + (D²³⁶ + D¹⁸¹ + 1)P(D) = 0period of 3 and a coding rate of ½ LDPC-CC Check polynomial #1: of atime (D³⁷³ + D⁵⁶ + 1)X₁(D) + (D⁸⁶ + D⁴ + 1)X₂(D) + (D⁴⁰⁶ + D²¹⁸ + 1)P(D)= 0 varying Check polynomial #2: period (D⁴⁵⁷ + D¹⁹⁷ + 1)X₁(D) + (D³⁶⁸ +D²⁹⁵ + 1)X₂(D) + (D⁴⁹¹ + D²² + 1)P(D) = 0 of 3 and Check polynomial #3:a coding (D⁴⁸⁵ + D⁷⁰ + 1)X₁(D) + (D⁴⁷⁵ + D³⁹⁸ + 1)X₂(D) + (D²³⁶ + D¹⁸¹ +1)P(D) = 0 rate of ⅔ LDPC-CC Check polynomial #1: (D³⁷³ + D⁵⁶ +1)X₁(D) + (D⁸⁶ + D⁴ + 1)X₂(D) + of a time (D³⁸⁸ + D¹³⁴ + 1)X₃(D) +(D⁴⁰⁶ + D²¹² + 1)P(D) = 0 varying Check polynomial #2: (D⁴⁵⁷ + D¹⁹⁷ +1)X₁(D) + (D³⁶⁸ + D²⁹⁵ + 1)X₂(D) + period (D¹⁵⁵ + D¹³⁶ + 1)X₃(D) +(D⁴⁹¹ + D⁷² + 1)P(D) = 0 of 3 and Check polynomial #3: (D⁴⁸⁵ + D⁷⁰ +1)X₁(D) + (D⁴⁷⁵ + D³⁹⁸ + 1)X₂(D) + a coding (D⁴⁹³ + D⁷⁷ + 1)X₃(D) +(D²³⁶ + D¹⁸¹ + 1)P(D) = 0 rate of ¾ LDPC-CC Check polynomial #1: of atime (D³⁷³ + D⁵⁶ + 1)X₁(D) + (D⁸⁶ + D⁴ + 1)X₂(D) + (D³⁸⁸ + D¹³⁴ +1)X₃(D) + varying (D²⁵⁰ + D¹⁹⁷ + 1)X₄(D) + (D²⁹⁵ + D¹¹³ + 1)X₅(D) +(D⁴⁰⁶ + D²¹⁸ + 1)P(D) = 0 period Check polynomial #2: of 3 and (D⁴⁵⁷ +D¹⁹⁷ + 1)X₁(D) + (D³⁶⁸ + D²⁹⁵ + 1)X₂(D) + (D¹⁵⁵ + D¹³⁶ + 1)X₃(D) + acoding (D²²⁰ + D¹⁴⁶ + 1)X₄(D) + (D³¹¹ + D¹¹⁵ + 1)X₅(D) + (D⁴⁹¹ + D²² +1)P(D) = 0 rate of Check polynomial #3: ⅚ (D⁴⁸⁵ + D⁷⁰ + 1)X₁(D) +(D⁴⁷⁵ + D³⁹⁸ + 1)X₂(D) + (D⁴⁹³ + D⁷⁷ + 1)X₃(D) + (D⁴⁹⁰ + D²³⁹ +1)X₄(D) + (D³⁹⁴ + D²⁷⁸ + 1)X₅(D) + (D²³⁶ + D¹⁸¹ + 1)P(D) = 0

A case will be explained where LDPC-CCs of a time varying period of 3 intable 5 satisfy the above conditions. For example, consider an LDPC-CCof a time varying period of 3 in a coding rate of ½ in table 5 and anLDPC-CC of a time varying period of 3 in a coding rate of ⅔ in table 5.That is, y=2 holds true in equations 31-1 to 31-g, and z=3 holds true inequations 32-1 to 32-g.

Then, seen from an LDPC-CC of a time varying period of 3 in a codingrate of ½ in table 5, A_(X1,0)(D) of equation 31-1 representsD³⁷³+D⁵⁶+1, and, seen from an LDPC-CC of a time varying period of 3 in acoding rate of ⅔ in table 5, A_(X1,0)(D) of equation 32-1 representsD³⁷³+D⁵⁶+1, so that A_(X1,0)(D) of equation 31-1 and A_(X1,0)(D) ofequation 32-1 are equal.

Also, seen from an LDPC-CC of a time varying period of 3 in a codingrate of ½ in table 5, B₀(D) of equation 31-1 represents D⁴⁰⁶+D²¹⁸+1 and,seen from an LDPC-CC of a time varying period of 3 in a coding rate of ⅔in table 5, B₀(D) of equation 32-1 represents D⁴⁰⁶+D²¹⁸+1, so that B₀(D)of equation 31-1 and B₀(D) of equation 32-1 are

Similarly, seen from an LDPC-CC of a time varying period of 3 in acoding rate of ½ in table 5, A_(X1,1)(D) of equation 31-2 representsD⁴⁵⁷+D¹⁹⁷+1, and, seen from an LDPC-CC of a time varying period of 3 ina coding rate of ⅔ in table 5, A_(X1,1)(D) of equation 32-2 representsD⁴⁵⁷+D¹⁹⁷+1, so that A_(X1,1)(D) of equation 31-2 and A_(X1,1)(D) ofequation 32-2 are equal.

Also, seen from an LDPC-CC of a time varying period of 3 in a codingrate of ½ in table 5, B₁(D) of equation 31-2 represents D⁴⁹¹+D²²+1, and,seen from an LDPC-CC of a time varying period of 3 in a coding rate of ⅔in table 5, B₁(D) of equation 32-2 represents D⁴⁹¹+D²²+1, so that B₁(D)of equation 31-2 and B₁(D) of equation 32-2 are equal.

Similarly, seen from an LDPC-CC of a time varying period of 3 in acoding rate of ½ in table 5, A_(X1,2)(D) of equation 31-3 representsD⁴⁸⁵+D⁷⁰+1, and, seen from an LDPC-CC of a time varying period of 3 in acoding rate of ⅔ in table 5, A_(X1,2)(D) of equation 32-3 representsD⁴⁸⁵+D⁷⁰+1, so that A_(X1,2)(D) of equation 31-3 and A_(X1,2)(D) ofequation 32-3 are equal.

Also, seen from an LDPC-CC of a time varying period of 3 in a codingrate of ½ in table 5, B₂(D) of equation 31-3 represents D²³⁶+D¹⁸¹+1,and, seen from an LDPC-CC of a time varying period of 3 in a coding rateof ⅔ in table 5, B₂(D) of equation 32-3 represents D²³⁶+D¹⁸¹+1, so thatB₂(D) of equation 31-3 and B₂(D) of equation 32-3 are equal.

In view of the above, it is confirmed that an LDPC-CC of a time varyingperiod of 3 in a coding rate of ½ in table 5 and an LDPC-CC of a timevarying period of 3 in a coding rate of ⅔ in table 5 satisfy the aboveconditions.

Similarly as above, if LDPC-CCs of a time varying period of 3 in twodifferent coding rates are selected from LDPC-CCs of a time varyingperiod of 3 in coding rates of ½, ⅔, ¾ and ⅚ in table 5, and whether ornot the above conditions are satisfied is examined, it is confirmed thatthe above conditions are satisfied in any selected patterns.

Also, an LDPC-CC is a class of a convolutional code, and thereforerequires, for example, termination or tail-biting to secure belief indecoding of information bits. Here, a case will be considered where themethod of making the state of data (information) X zero (hereinafter“information-zero-termination”) is implemented.

FIG. 10 shows the method of information-zero-termination. As shown inFIG. 10, the information bit (final transmission bit) that is finallytransmitted in a transmission information sequence is Xn(110). With thisfinal information bit Xn(110), if the transmitting apparatus transmitsdata only until parity bits generated in an encoder and then thereceiving apparatus performs decoding, the received quality ofinformation degrades significantly. To solve this problem, informationbits subsequent to final information bit Xn(110) (hereinafter “virtualinformation bits”) are presumed as “0” and encoded to generate paritybits 130.

In this case, the receiving apparatus knows that virtual informationbits 120 are “0,” so that the transmitting apparatus does not transmitvirtual information bits 120, but transmits only parity bits 130generated by virtual information bits 120 (these parity bits representredundant bits that need to be transmitted, and therefore are called“redundant bits”). Then, a new problem arises that, in order to enableboth improvement of efficiency of data transmission and maintenance ofreceived quality of data, it is necessary to secure the received qualityof data and decrease the number of parity bits 130 generated by virtualinformation bits 120 as much as possible.

At this time, it is confirmed by simulation that, in order to secure thereceived quality of data and decrease the number of parity bitsgenerated by virtual information bits, terms related to parity of aparity check polynomial play an important role.

As an example, a case will be explained using an LDPC-CC for which thetime varying period is m (where m is an integer equal to or greater than2) and the coding rate is ½. When the time varying period is m, mnecessary parity check polynomials are represented by the followingequation.[33]A _(X1,1)(D)X ₁(D)+B ₁(D)P(D)=0  (Equation 33)

where i=0, 1, . . . , m−1. Also, assume that all of the orders of D inA_(X1,1)(D) are integers equal to or greater than 0 (e.g. as shown inA_(X1,1)(D)=D¹⁵+D³D⁰, the orders of D are 15, 3 and 0, all of which areintegers equal to or greater than 0), and all of the orders of D inB₁(D) are also integers equal to or greater than 0 (e.g. as shown inB₁(D)=D¹⁸+D⁴D⁰, the orders of D are 18, 4 and 0, all of which areintegers equal to or greater than 0).

Here, at time j, the parity check polynomial of the following equationholds true.[34]A _(X1,k)(D)X ₁(D)+B _(k)(D)P(D)=0 (k=j mod m)  (Equation 34)

Then, in X₁(D), assume that: the highest order of D in A_(X1,1)(D) is α₁(e.g. when A_(X1,1)(D)=D¹⁵+D³+D⁰, D has the orders of 15, 3 and 0, andtherefore provides 15 as the highest order of D, α₁); the highest orderof D in A_(X1,2)(D) is α₂; . . . ; the highest order of D in A_(X1,1)(D)is α_(i); . . . ; and the highest order of D in A_(X1,m-1)(D) isα_(m-1). Then, the highest value in α_(i) (where i=0, 1, 2, . . . , m−1)is α.

On the other hand, in P(D), assume that: the highest order of D in B₁(D)is β₁; the highest order of D in B₂(D) is β₂; . . . ; the highest orderof D in B_(i)(D) is β_(i); . . . ; and the highest order of D inB_(m-1)(D) is β_(m-1). Then, the highest value in β_(i) (where i=0, 1,2, . . . , m−1) is β.

Then, in order to secure the received quality of data and decrease thenumber of parity bits generated by virtual information bits as much aspossible, it is preferable to set β equal to or below half of α.

Although a case has been described where the coding rate is ½, the sameapplies to other cases where the coding rate is above ½. At this time,especially when the coding rate is equal to or greater than ⅘, there isa trend to require a significant number of redundant bits to satisfyconditions for securing the received quality of data and decreasing thenumber of parity bits generated by virtual information bits as much aspossible. Consequently, the conditions described above play an importantrole to secure the received quality of data and decrease the number ofparity bits generated by virtual information bits as much as possible.

As an example, a case will be explained using an LDPC-CC for which thetime varying period is m (where m is an integer equal to or greater than2) and the coding rate is ⅘. When the time varying period is m, mnecessary parity check polynomials are represented by the followingequation.[35]A _(X1,i)(D)X ₁(D)+A _(X2,i)(D)X ₂(D)+A _(X3,i)(D)X ₃(D)+A _(X4,i)(D)X₄(D)+B _(i)(D)P(D)=0  (Equation 35)

where i=0, 1, . . . , m−1. Also, assume that all of the orders of D inA_(X1,i)(D) are integers equal to or greater than 0 (e.g. as shown inA_(X1,1)(D)=D¹⁵+D³+D⁰, the orders of D are 15, 3 and 0, all of which areintegers equal to or greater than 0). Similarly, assume that: all of theorders of D in A_(X1,1)(D) are integers equal to or greater than 0; allof the orders of D in A_(X3,i)(D) are integers equal to or greater than0; all of the orders of D in A_(X4,i)(D) are integers equal to orgreater than 0; and all of the orders of D in B_(i)(D) are integersequal to or greater than 0 (e.g. as shown in B_(i)(D)=D¹⁸+D⁴+D⁰, theorders of D are 18, 4 and 0, all of which are integers equal to orgreater than 0).

Here, at time j, the parity check polynomial of the following equationholds true.[36]A _(X1,k)(D)X ₁(D)+A _(X2,k)(D)X ₂(D)+A _(X3,k)(D)X ₃(D)+A _(X4,k)(D)X_(4i)(D)+B _(k)(D)P(D)=0 (k=j mod m)  (Equation 36)

Then, in X₁(D), assume that: the highest order of D in A_(X1,1)(D) isα_(1,1) (e.g. when A_(X1,1)(D)=D¹⁵+D³+D⁰, D has the orders of 15, 3 and0, and therefore provides 15 as the highest order of D, α_(1,1)); thehighest order of D in A_(X1,2)(D) is α_(1,2); . . . ; the highest orderof D in A_(X1,i)(D) is α_(1,1); . . . ; and the highest order of D inA_(X1,m-1)(D) is α_(1,m-1). Then, the highest value in α_(1,i) (wherei=0, 1, 2, . . . , m−1) is α₁.

In X₂(D), assume that: the highest order of D in A_(X2,1)(D) is α_(2,1)(e.g. when A_(X2,1)(D)=D¹⁵+D³+D⁰, D has the orders of 15, 3 and 0, andtherefore provides 15 as the highest order of D, α_(2,1)); the highestorder of D in A_(X2,2)(D) is α_(2,2); . . . ; the highest order of D inA_(X2,i)(D) is αt_(2,i); . . . ; and the highest order of D inA_(X2,m-1)(D) is α_(2, m-1). Then, the highest value in α_(2,i) (wherei=0, 1, 2, . . . , m−1) is α₂.

In X₃(D), assume that: the highest order of D in A_(X3,1)(D) is α_(3,1)(e.g. when A_(X3,1)(D)=D¹⁵+D³+D⁰, D has the orders of 15, 3 and 0, andtherefore provides 15 as the highest order of D, α_(3,1)); the highestorder of D in A_(X3,2)(D) is α_(3,2); . . . ; the highest order of D inA_(X3,i)(D) is α_(3,i); . . . ; and the highest order of D inA_(X3,m-1)(D) is α_(3,m-1). Then, the highest value in α_(3,1) (wherei=0, 1; 2, . . . , m−1) is α₃.

In X₄(D), assume that: the highest order of D in A_(X4,1)(D) is α_(4,1)(e.g. when A_(X4,1)(D)=D¹⁵+D³+D⁰, D has the orders of 15, 3 and 0, andtherefore provides 15 as the highest order of D, α_(4,1)); the highestorder of D in A_(X4,2)(D) is α_(4,2); . . . ; the highest order of D inA_(X4,i)(D) is α_(4,i); . . . ; and the highest order of D inA_(X4,m-1)(D) is α_(4,m-1). Then, the highest value in α_(4,i) (wherei=0, 1, 2, . . . , m−1) is α₄.

In P(D), assume that: the highest order of D in B₁(D) is β₁; the highestorder of D in B₂(D) is β₂; . . . ; the highest order of D in B_(i)(D) isβ_(i); . . . ; and the highest order of D in B_(m-1)(D) is β_(m-1).Then, the highest value in β_(i) (where i=0, 1, 2, . . . , m−1) is β.

Then, in order to secure the received quality of data and decrease thenumber of parity bits generated by virtual information bits as much aspossible, it is necessary to satisfy conditions that: β is equal to orbelow half of α₁; β is equal to or below half of α₂; β is equal to orbelow half of α₃; and β is equal to or below half of α₄, so that,especially, there is a high possibility to secure the received qualityof data.

Also, even in a case where: β is equal to or below half of α₁; β isequal to or below half of α₂; β is equal to or below half of α₃; or β isequal to or below half of α₄, although it is possible to secure thereceived quality of data and decrease the number of parity bitsgenerated by virtual information bits as much as possible, there is alittle possibility to cause degradation in the received quality of data(here, degradation in the received quality of data is not necessarilycaused).

Therefore, in the case of an LDPC-CC for which the time varying periodis m (where m is an integer equal to or greater than 2) and the codingrate is (n−1)/n, the following is possible.

When the time varying period is m, m necessary parity check polynomialsare represented by the following equation.[37]A _(X1,i)(D)X ₁(D)+A _(X2,i)(D)X ₂(D)+ . . . +A _(Xn-1,i)(D)X_(n-1)(D)+B _(i)(D)P(D)=0  (Equation 37)

where i=0, 1, . . . , m−1. Also, assume that all of the orders of D inA_(X1,i)(D) are integers equal to or greater than 0 (e.g. as shown inA_(X1,1)(D)=D¹⁵+D³+D⁰, the orders of D are 15, 3 and 0, all of which areintegers equal to or greater than 0). Similarly, assume that: all of theorders of D in A_(X2,i)(D) are integers equal to or greater than 0; allof the orders of D in A_(X3,i)(D) are integers equal to or greater than0; all of the orders of D in A_(X4,i)(D) are integers equal to orgreater than 0; . . . ; all of the orders of D in A_(Xα,i)(D) areintegers equal to or greater than 0; . . . ; all of the orders of D inA_(Xn-1,i)(D) are integers equal to or greater than 0; and all of theorders of D in B_(i)(D) are integers equal to or greater than 0 (e.g. asshown in B_(i)(D)=D¹⁸+D⁴+D⁰, the orders of D are 18, 4 and 0, all ofwhich are integers equal to or greater than 0).

Here, at time j, the parity check polynomial of the following equationholds true.[38]A _(X1,k)(D)X ₁(D)+A _(X2,k)(D)X ₂(D)+ . . . +A _(Xn-1,k)(D)X_(n-1)(D)+B _(k)(D)P(D)=0 (k=j mod m)  (Equation 38)

Then, in X₁(D), assume that: the highest order of D in A_(X1,1)(D) isα_(1,1) (e.g. when A_(X1,1)(D)=D¹⁵+D³+D⁰, D has the orders of 15, 3 and0, and therefore provides 15 as the highest order of D, α_(1,1)); thehighest order of D in A_(X1,2)(D) is α_(1,2); . . . ; the highest orderof D in A_(X1,i)(D) is α_(1,i); . . . ; and the highest order of D inA_(X1,m-1)(D) is α_(1,m-1). Then, the highest value in α_(1,i) (wherei=0, 1, 2, . . . , m−1) is α₁.

In X₂(D), assume that: the highest order of D in A_(X2,1)(D) is α_(2,1)(e.g. when A_(X2,1)(D)=D¹⁵+D³+D⁰, D has the orders of 15, 3 and 0, andtherefore provides 15 as the highest order of D, α_(2,1)); the highestorder of D in A_(X2,2)(D) is α_(2,2); . . . ; the highest order of D inA_(X2,i)(D) is α_(2,i); . . . ; and the highest order of D inA_(X2,m-1)(D) is α_(2,m-1). Then, the highest value in α_(2,1) (wherei=0, 1, 2, . . . , m−1) is α₂.

In X_(u)(D), assume that: the highest order of D in A_(Xu,1)(D) isα_(u,1) (e.g. when A_(Xu,1)(D)=D¹⁵+D³+D⁰, D has the orders of 15, 3 and0, and therefore provides 15 as the highest order of D, α_(u,1)); thehighest order of D in A_(Xu,2)(D) is α_(u,2); . . . ; the highest orderof D in A_(Xu,i)(D) is α_(u,i); . . . ; and the highest order of D inA_(Xu,m-1)(D) is α_(u,m-1). Then, the highest value in α_(u,i) (wherei=0, 1, 2, . . . , m−1, u=1, 2, 3, . . . , n−2, n−1) is α_(u).

-   -   . . .

In X_(n-1)(D), assume that: the highest order of D in A_(Xn-1,1)(D) isα_(n-1,1) (e.g. when A_(Xn-1,1)(D)=D¹⁵+D³+D⁰, D has the orders of 15, 3and 0, and therefore provides 15 as the highest order of D, α_(n-1,1));the highest order of D in A_(Xn-1,2)(D) is α_(n-1,2); . . . ; thehighest order of D in A_(Xn-1,i)(D) is α_(n-1,i); . . . ; and thehighest order of D in A_(Xn-1,m-1)(D) is α_(n-1,m-1). Then, the highestvalue in α_(n-1,i) (where i=0, 1, 2, . . . , m−1) is α_(n-1).

In P(D), assume that: the highest order of D in B₁(D) is β₁; the highestorder of D in B₂(D) is β₂; . . . ; the highest order of D in B_(i)(D) isβ_(i); . . . ; and the highest order of D in B_(m-1)(D) is β_(m-1).Then, the highest value in β_(i) (where i=0, 1, 2, . . . , m−1) is β.

Then, in order to secure the received quality of data and decrease thenumber of parity bits generated by virtual information bits as much aspossible, it is necessary to satisfy conditions that: β is equal to orbelow half of α₁; β is equal to or below half of α₂; . . . ; β is equalto or below half of α_(u); . . . ; and β is equal to or below half ofα_(n-1) (where u=1, 2, 3, . . . , n−2, n−1), so that, especially, thereis a high possibility to secure the received quality of data.

Also, even in a case where: β is equal to or below half of α₁; β isequal to or below half of α₂; . . . ; β is equal to or below half ofα_(u); . . . ; or β is equal to or below half of α_(n-1) (where u=1, 2,3, . . . , n−2, n−1), although it is possible to secure the receivedquality of data and decrease the number of parity bits generated byvirtual information bits as much as possible, there is a littlepossibility to cause degradation in the received quality of data (here,degradation in the received quality of data is not necessarily caused).

Table 6 shows an example of LDCPC-CC parity check polynomials that cansecure the received quality of data and reduce redundant bits, where thetime varying period is 3 and the coding rate is ½, ⅔, ¾ or ⅘. IfLDPC-CCs of a time varying period of 3 in two different coding rates areselected from LDPC-CCs of a time varying period of 3 in coding rates of½, ⅔, ¾ and ⅚ in table 6, and whether or not the above-describedconditions for sharing encoders and decoders are satisfied is examined,similar to LDPC-CCs of a time varying period of 3 in table 5, it isconfirmed that the above conditions for enabling sharing process inencoders and decoders are satisfied in any selected patterns.

Also, although 1000 redundant bits are required in a coding rate of ⅚ intable 5, it is confirmed that the number of redundant bits is equal toor below 500 bits in a coding rate of ⅘ in table 6.

Also, in the codes of table 6, the number of redundant bits (which areadded for information-zero-termination) varies between coding rates. Atthis time, the number of redundant bits tends to increase when thecoding rate increases. However, this tendency is not always seen.Furthermore, when the coding rate is large and the information size islarge, the number of redundant bits tends to increase. That is, whencodes are created as shown in Table 5 and Table 6, if there are a codeof a coding rate of (n−1)/n and a code of a coding rate of (m−1)/m(n>m), the number of redundant bits necessary for the code of a codingrate of (n−1)/n (redundant bits added for“information-zero-termination”) tends to be greater than the number ofredundant bits necessary for the code of a coding rate of (m−1)/m(redundant bits added for “information-zero-termination”), and moreoverwhen the information size is small, the number of redundant bitsnecessary for the code of a coding rate of (n−1)/n tends to be greaterthan the number of redundant bits necessary for the code of a codingrate of (m−1)/m. However, such a tendency is not always observed.

TABLE 6 Code Parity check polynomial LDPC-CC Check polynomial #1:(D²⁶⁸ + D¹⁶⁴ + 1)X₁(D) + (D⁹² + D⁷ + 1)P(D) = 0 of a time Checkpolynomial #2: (D³⁷⁰ + D³¹⁷ + 1)X₁(D) + (D⁹⁵ + D²² + 1)P(D) = 0 varyingCheck polynomial #3: (D³⁴⁶ + D⁸⁶ + 1)X₁(D) + (D⁸² + D²⁶ + 1)P(D) = 0period of 3 and a coding rate of ½ LDPC-CC Check polynomial #1: of atime (D²⁶⁸ + D¹⁶⁴ + 1)X₁(D) + (D³⁸⁵ + D²⁴² + 1)X₂(D) + (D⁹² + D⁷ +1)P(D) = 0 varying Check polynomial #2: period (D³⁷⁰ + D³¹⁷ + 1)X₁(D) +(D¹²⁵ + D¹⁰³ + 1)X₂(D) + (D⁹⁵ + D²² + 1)P(D) = 0 of 3 and Checkpolynomial #3: a coding (D³⁴⁶ + D⁸⁶ + 1)X₁(D) + (D³¹⁹ + D²⁹⁰ + 1)X₂(D) +(D⁸⁸ + D²⁶ + 1)P(D) = 0 rate of ⅔ LDPC-CC Check polynomial #1: (D²⁶⁸ +D¹⁶⁴ + 1)X₁(D) + (D³⁸⁵ + D²⁴² + 1)X₂(D) + of a time (D³⁴³ + D²⁸⁴ +1)X₃(D) + (D⁹² + D⁷ + 1)P(D) = 0 varying Check polynomial #2: (D³⁷⁰ +D³¹⁷ + 1)X₁(D) + (D¹²⁵ + D¹⁰³ + 1)X₂(D) + period (D²⁵⁹ + D¹⁴ + 1)X₃(D) +(D⁹⁵ + D²² + 1)P(D) = 0 of 3 and Check polynomial #3: (D³⁴⁶ + D⁸⁶ +1)X₁(D) + (D³¹⁹ + D²⁹⁰ + 1)X₂(D) + a coding (D¹⁴⁵ + D¹¹ + 1)X₃(D) +(D⁸⁸ + D²⁶ + 1)P(D) = 0 rate of ¾ LDPC-CC Check polynomial #1: of a time(D²⁶⁸ + D¹⁶⁴ + 1)X₁(D) + (D³⁸⁵ + D²⁴² + 1)X₂(D) + varying (D³⁴³ + D²⁸⁴ +1)X₃(D) + (D³¹⁰ + D¹¹³ + 1)X₄(D) + (D⁹² + D⁷ + 1)P(D) = 0 period Checkpolynomial #2: of 3 and (D³⁷⁰ + D³¹⁷ + 1)X₁(D) + (D¹²⁵ + D¹⁰³ +1)X₂(D) + a coding (D²⁵⁹ + D¹⁴ + 1)X₃(D) + (D³⁹⁴ + D¹⁸⁸ + 1)X₄(D) +(D⁹⁵ + D²² + 1)P(D) = 0 rate of Check polynomial #3: ⅘ (D³⁴⁶ + D⁸⁶ +1)X₁(D) + (D³¹⁹ + D²⁹⁰ + 1)X₂(D) + (D¹⁴⁵ + D¹¹ + 1)X₃(D) + (D²³⁹ + D⁶⁷ +1)X₄(D) + (D⁸⁸ + D²⁶ + 1)P(D) = 0

A case has been described above where the maximum coding rate is (q−1)/qamong coding rates of enabling encoder circuits to be shared andenabling decoder circuits to be shared, and where an LDPC-CC paritycheck polynomial of a coding rate of (r−1)/r (r=2, 3, . . . , q (q is anatural number equal to or greater than 3)) and a time varying period ofg is provided.

Here, the method of generating an LDPC-CC parity check polynomial of atime varying period of g for reducing the computational complexity (i.e.circuit scale) in a transmitting apparatus and receiving apparatus, andfeatures of parity check polynomials have been described, where thetransmitting apparatus provides at least an LDPC-CC encoder of a codingrate of (y−1)/y and a time varying period of g and an LDPC-CC encoder ofa coding rate of (z−1)/z (y≠z) and a time varying period of g, and wherethe receiving apparatus provides at least an LDPC-CC decoder of a codingrate of (y−1)/y and a time varying period of g and an LDPC−CC decoder ofa coding rate of (z−1)/z (y≠z) and a time varying period of g.

Here, the transmitting apparatus refers to a transmitting apparatus thatcan generate at least one of a modulation signal for transmitting anLDPC-CC coding sequence of a coding rate of (y−1)/y and a time varyingperiod of g and an LDPC-CC coding sequence of a coding rate of (z−1)/zand a time varying period of g.

Also, the receiving apparatus refers to a receiving apparatus thatdemodulates and decodes at least one of a received signal including anLDPC-CC coding sequence of a coding rate of (y−1)/y and a time varyingperiod of g and a received signal including an LDPC-CC coding sequenceof a coding rate of (z−1)/z and a time varying period of g.

By using an LDPC-CC of a time varying period of g proposed by thepresent invention, it is possible to provide an advantage of reducingthe computational complexity (i.e. circuit scale) in a transmittingapparatus including encoders and in a receiving apparatus includingdecoders (i.e. it is possible to share circuits).

Further, by using an LDPC-CC of a time varying period of g proposed bythe present invention, it is possible to provide an advantage ofacquiring data of high received quality in the receiving apparatus inany coding rates. Also, the configurations of encoders and decoders, andtheir operations will be described later in detail.

Also, although a case has been described above where LDPC-CCs of a timevarying period of g in coding rates of ½, ⅔, ¾, . . . , and (q−1)/q areprovided in equations 30-1 to 30-(q−1), a transmitting apparatusincluding encoders and a receiving apparatus including decoders need notsupport all of the coding rates of ½, ⅔, ¾, . . . , and (q−1)/q. Thatis, as long as these apparatuses support at least two or more differentcoding rates, it is possible to provide an advantage of reducing thecomputational complexity (or circuit scale) in the transmittingapparatus and the receiving apparatus (i.e. sharing encoder circuits anddecoder circuits), and acquiring data of high received quality in thereceiving apparatus.

Also, if all of coding rates supported by the transmitting and receivingapparatuses (encoders/decoders) are associated with codes based on themethods described with the present embodiment, by providingencoders/decoders of the highest coding rate among the supported codingrates, it is possible to easily support coding and decoding in allcoding rates and, at this time, provide an advantage of reducing thecomputational complexity significantly.

Furthermore, although the present embodiment has been described based onthe code (LDPC-CC with good characteristics) described in Embodiment 1,the above-described condition (LDPC-CC with good characteristics) neednot always be satisfied, and the present embodiment can be likewiseimplemented if it is an LDPC-CC of a time varying period of g (g is aninteger equal to or greater than 2) based on a parity check polynomialof the above-described format (LDPC-CC with good characteristics). Thisis obvious from the relationships between equations 31-1 to 31-g andequations 32-1 to 32-g.

Naturally, for example, in a case where: the transmitting and receivingapparatuses (encoders/decoders) support coding rates of ½, ⅔, ¾ and ⅚;LDPC-CCs based on the above conditions are used in coding rates of ½, ⅔and ¾; and codes not based on the above conditions are used in a codingrate of ⅚, it is possible to share circuits in the encoders and decodersin coding rates of ½, ⅔ and ¾, and it is difficult to share circuits inthese encoders and decoders to share circuits in a coding rate of ⅚.

Embodiment 3

The present embodiment will explain in detail the method of sharingencoder circuits of an LDPC-CC formed by the search method explained inEmbodiment 2 and the method of sharing decoder circuits of that LDPC-CC.

First, in a case where the highest coding rate is (q−1)/q among codingrates for sharing encoder circuits and decoder circuits, an LDPC-CCencoder of a time varying rate of g (where g is a natural number)supporting a plurality of coding rates, (r−1)/r, will be explained (e.g.when the coding rates supported by a transmitting and receivingapparatus are ½, ⅔, ¾ and ⅚, coding rates of ½, ⅔ and ¾ allow thecircuits of encoders/decoders to be shared, while a coding rate of ⅚does not allow the circuits of encoders/decoders to be shared, where theabove highest coding rate, (q−1)/q, is ¾).

FIG. 11 is a block diagram showing an example of the main components ofan encoder according to the present embodiment. Also, encoder 200 shownin FIG. 11 refers to an encoder that can support coding rates of ½, ⅔and ¾. Encoder 200 shown in FIG. 11 is mainly provided with informationgenerating section 210, first information computing section 220-1,second information computing section 220-2, third information computingsection 220-3, parity computing section 230, adding section 240, codingrate setting section 250 and weight control section 260

Information generating section 210 sets information X_(1,i), informationX_(2,i) and information X_(3,i) at point in time i, according to acoding rate designated from coding rate setting section 250. Forexample, if coding rate setting section 250 sets the coding rate to ½,information generating section 210 sets information X_(1,i) at point intime i to input information data S_(j), and sets information X_(2,i) andinformation X_(3,i) at point in time i to “0.”

Also, in the case of a coding rate of ⅔, information generating section210 sets information X_(1,i) at point in time i to input informationdata S_(j), sets information X_(2,i) at point in time i to inputinformation data S_(j+1) and sets information X_(3,i) at point in time ito “0.”

Also, in the case of a coding rate of ¾, information generating section210 sets information X_(1,i) at point in time i to input informationdata S_(j), sets information X_(2,i) at point in time i to inputinformation data S_(j+1) and sets information X_(3,i) at point in time ito input information data S_(j+2).

In this way, using input information data, information generatingsection 210 sets information X_(1,i), information X_(2,i) andinformation X_(3,i) at point in time i according to a coding rate set incoding rate setting section 250, outputs set information X_(1,i) tofirst information computing section 220-1, outputs set informationX_(2,i) to second information computing section 220-2 and outputs setinformation X_(3,i) to third information computing section 220-3.

First information computing section 220-1 calculates X₁(D) according toA_(X1,k)(D) of equation 30-1. Similarly, second information computingsection 220-2 calculates X₂(D) according to A_(X2,k)(D) of equation30-2. Similarly, third information computing section 220-3 calculatesX₃(D) according to A_(X3,k)(D) of equation 30-3.

At this time, as described in Embodiment 2, from the conditions tosatisfy in equations 31-1 to 31-g and 32-1 to 32-g, if the coding rateis changed, it is not necessary to change the configuration of firstinformation computing section 220-1, and, similarly, change theconfiguration of second information computing section 220-2 and changethe configuration of third information computing section 220-3.

Therefore, when a plurality of coding rates are supported, by using theconfiguration of the encoder of the highest coding rate as a referenceamong coding rates for sharing encoder circuits, the other coding ratescan be supported by the above operations. That is, regardless of codingrates, LDPC-CCs explained in Embodiment 2 provide an advantage ofsharing first information computing section 220-1, second informationcomputing section 220-2 and third information computing section 220-3,which are main components of the encoder. Also, for example, theLDPC-CCs shown in table 5 provides an advantage of providing data ofgood received quality regardless of coding rates.

FIG. 12 shows the configuration inside first information computingsection 220-1. First information computing section 220-1 in FIG. 12 isprovided with shift registers 221-1 to 221-M, weight multipliers 220-0to 222-M and adder 223.

Shift registers 222-1 to 222-M are registers each storing X_(1,i-t)(where t=0, . . . , M), and, at a timing at which the next input comesin, send a stored value to the adjacent shift register to the right, andstore a value sent from the adjacent shift register to the left.

Weight multipliers 220-0 to 222-M switch a value of h₁ ^((m)) to 0 or 1in accordance with a control signal outputted from weight controlsection 260.

Adder 223 performs exclusive OR computation of outputs of weightmultipliers 222-0 to 222-M to find and output computation result Y_(1,i)to adder 240 in FIG. 11.

Also, the configurations inside second information computing section220-2 and third information computing section 220-3 are the same asfirst information computing section 220-1, and therefore theirexplanation will be omitted. In the same way as in first informationcomputing section 220-1, second information computing section 220-2finds and outputs calculation result Y_(2,i) to adder 240. In the sameway as in first information computing section 220-1, third informationcomputing section 220-3 finds and outputs calculation result Y_(3,i) toadder 240 in FIG. 11.

Parity computing section 230 in FIG. 11 calculates P(D) according toB_(k)(D) of equations 30-1 to 30-3.

Parity computing section 230 in FIG. 11 calculates P(D) according toB_(k)(D) of equations 30-1 to 30-3.

FIG. 13 shows the configuration inside parity computing section 230 inFIG. 11. Parity computing section 230 in FIG. 13 is provided with shiftregisters 231-1 to 231-M, weight multipliers 232-0 to 232-M and adder233.

Shift registers 231-1 to 231-M are registers each storing P_(i-t) (wheret=0, . . . , M), and, at a timing at which the next input comes in, senda stored value to the adjacent shift register to the right, and store avalue sent from the adjacent shift register to the left.

Weight multipliers 232-0 to 232-M switch a value of h₂ ^((m)) to 0 or 1in accordance with a control signal outputted from weight controlsection 260.

Adder 223 performs exclusive OR computation of outputs of weightmultipliers 232-0 to 232-M to find and output computation result Z_(i)to adder 240 in FIG. 11.

Referring back to FIG. 11 again, adder 240 performs exclusive ORcomputation of computation results Y_(1,i), Y_(2,i), Y_(3,i) and Z_(i)outputted from first information computing section 220-1, secondinformation computing section 220-2, third information computing section220-3 and parity computing section 230, to find and output parity P_(i)at point in time i. Adder 240 also outputs parity P_(i) at point in timei to parity computing section 230.

Coding rate setting section 250 sets the coding rate of encoder 200 andoutputs coding rate information to information generating section 210.

Based on a parity check matrix supporting equations 30-1 to 30-3 held inweight control section 260, weight control section 260 outputs the valueof h₁ ^((m)) at point in time i based on the parity check polynomials ofequations 30-1 to 30-3, to first information computing section 220-1,second information computing section 220-2, third information computingsection 220-3 and parity computing section 230. Also, based on theparity check matrix supporting equations 30-1 to 30-3 held in weightcontrol section 260, weight control section 260 outputs the value of h₂^((m)) at that timing to weight multipliers 232-0 to 232-M.

Also, FIG. 14 shows another configuration of an encoder according to thepresent embodiment. In the encoder of FIG. 14, the same components as inthe encoder of MGM are assigned the same reference numerals. Encoder 200of FIG. 14 differs from encoder 200 of FIG. 11 in that coding ratesetting section 250 outputs coding rate information to first informationcomputing section 220-1, second information computing section 220-2,third information computing section 220-3 and parity computing section230.

In the case where the coding rate is ½, second information computingsection 220-2 outputs “0” to adder 240 as computation result Y_(2,i),without computation processing. Also, in the case where the coding rateis ½ or ⅔, third information computing section 220-3 outputs “0” toadder 240 as computation result Y_(3,i), without computation processing.

Here, although information generating section 210 of encoder 200 in FIG.11 sets information X_(2,i) and information X_(3,i) at point in time ito “0” according to a coding rate, second information computing section220-2 and third information computing section 220-3 of encoder 200 inFIG. 14 stop computation processing according to a coding rate andoutput 0 as computation results Y_(2,i) and Y_(3,i). Therefore, theresulting computation results in encoder 200 of FIG. 14 are the same asin encoder 200 of FIG. 11.

Thus, in encoder 200 of FIG. 14, second information computing section220-2 and third information computing section 220-3 stops computationprocessing according to a coding rate, so that it is possible to reducecomputation processing, compared to encoder 200 of FIG. 11.

Next, the method of sharing LDPC-CC decoder circuits described inEmbodiment 2 will be explained in detail.

FIG. 15 is a block diagram showing the main components of a decoderaccording to the present embodiment. Here, decoder 300 shown in FIG. 15refers to a decoder that can support coding rates of ½, ⅔ and ¾. Decoder300 of FIG. 14 is mainly provided with log likelihood ratio settingsection 310 and matrix processing computing section 320.

Log likelihood ratio setting section 310 receives as input a receptionlog likelihood ratio and coding rate calculated in a log likelihoodratio computing section (not shown), and inserts a known log likelihoodratio in the reception log likelihood ratio according to the codingrate.

For example, when the coding rate is ½, it means that encoder 200transmits “0” as X_(2,i) and X_(3,i), and, consequently, log likelihoodratio setting section 310 inserts a fixed log likelihood ratio for theknown bit “0” as the log likelihood ratios of X_(2,i) and X_(3,i), andoutputs the inserted log likelihood ratios to matrix processingcomputing section 320. This will be explained below using FIG. 16.

As shown in FIG. 16, when the coding rate is ½, log likelihood ratiosetting section 310 receives reception log likelihood ratios LLR_(X1,i)and LLR_(Pi) corresponding to X_(1,i) and P_(i), respectively.Therefore, log likelihood ratio setting section 310 inserts receptionlog likelihood ratios LLR_(X2,i) and LLR_(3,i) corresponding to X_(2,i)and X_(3,i), respectively. In FIG. 16, reception log likelihood ratioscircled by doted lines represent reception log likelihood ratiosLLR_(X2,i) and LLR_(3,i) inserted by log likelihood ratio settingsection 310. Log likelihood ratio setting section 310 insertsfixed-value log likelihood ratios as reception log likelihood ratiosLLR_(X2,i) and LLR_(3,i).

Also, in the case where the coding rate is ⅔, it means that encoder 200transmits “0” as X_(3,i), and, consequently, log likelihood ratiosetting section 310 inserts a fixed log likelihood ratio for the knownbit “0” as the log likelihood ratio of X_(3,i), and outputs the insertedlog likelihood ratio to matrix processing computing section 320. Thiswill be explained using FIG. 17.

As shown in FIG. 17, in the case where the coding rate is ⅔, loglikelihood ratio setting section 310 receives as input reception loglikelihood ratios LLR_(X1), LLR_(x2,i) and LLR_(Pi) corresponding toX_(1,i), X_(2,i) and P_(i), respectively. Therefore, log likelihoodratio setting section 310 inserts reception log likelihood ratioLLR_(3,i) corresponding to X_(3,i). In FIG. 17, reception log likelihoodratios circled by doted lines represent reception log likelihood ratioLLR_(3,i) inserted by log likelihood ratio setting section 310. Loglikelihood ratio setting section 310 inserts fixed-value log likelihoodratios as reception log likelihood ratio LLR_(3,i).

Matrix processing computing section 320 in FIG. 15 is provided withstorage section 321, row processing computing section 322 and columnprocessing computing section 323.

Storage section 321 stores a log likelihood ratio, external value α_(mn)obtained by row processing and a priori value β_(mn) obtained by columnprocessing.

Row processing computing section 322 holds the row-direction weightpattern of LDPC-CC parity check matrix H of the maximum coding rate of ¾among coding rates supported by encoder 200. Row processing computingsection 322 reads a necessary priori value β_(mn) from storage section321, according to that row-direction weight pattern, and performs rowprocessing computation.

In row processing computation, row processing computation section 322decodes a single parity check code using a priori value β_(mn), andfinds external value α_(mn).

Processing of the m-th row will be explained. Here, an LDPC code paritycheck matrix to decode two-dimensional M×N matrix H={H_(mn)} will beused. External value α_(mn) is updated using the following updateequation for all pairs (m, n) satisfying the equation H_(mn)=1.

$\begin{matrix}\lbrack 39\rbrack & \; \\{\alpha_{m\; n} = {( {\prod\limits_{n^{\prime} \in {{A{(m)}}\backslash\; n}}{{sign}( \beta_{m\; n^{\prime}} )}} ){\Phi( {\sum\limits_{n^{\prime} \in {{A{(m)}}\backslash\; n}}{\Phi( {\beta_{m\; n^{\prime}}} )}} )}}} & ( {{Equation}\mspace{14mu} 39} )\end{matrix}$

where Φ(x) is called a Gallager f function, and is defined by thefollowing equation.

$\begin{matrix}\lbrack 40\rbrack & \; \\{{\Phi(x)} = {\ln\frac{{\exp(x)} + 1}{{\exp(x)} - 1}}} & ( {{Equation}\mspace{14mu} 40} )\end{matrix}$

Column processing computing section 323 holds the column-directionweight pattern of LDPC-CC parity check matrix H of the maximum codingrate of ¾ among coding rates supported by encoder 200. Column processingcomputing section 323 reads a necessary external value α_(mn) fromstorage section 321, according to that column-direction weight pattern,and finds a priori value β_(mn).

In column processing computation, column processing computing section323 performs iterative decoding using input log likelihood ratio λ_(n)and external value α_(mn), and finds a priori value β_(mn).

Processing of the m-th column will be explained. β_(mn) is updated usingthe following update equation for all pairs (m, n) satisfying theequation H_(mn)=1. Only when q=1, the calculation is performed withα_(mn)=0.

$\begin{matrix}\lbrack 41\rbrack & \; \\{\beta_{m\; n} = {\lambda_{n} + {\sum\limits_{m^{\prime} \in {{B{(n)}}/m}}\alpha_{m^{\prime}n}}}} & ( {{Equation}\mspace{14mu} 41} )\end{matrix}$

After repeating above row processing and column processing apredetermined number of times, decoder 300 finds an a posteriori loglikelihood ratio.

As described above, with the present embodiment, in a case where thehighest coding rate is (q−1)/q among supported coding rates and wherecoding rate setting section 250 sets the coding rate to (s−1)/s,information generating section 310 sets from information X_(s,i) toinformation X_(q-1,i) as “0.” For example, when supported coding ratesare ½, ⅔ and ¾ (q=4), first information computing section 220-1 receivesas input information X_(1,i) at point in time i and calculates termX₁(D) of equation 30-1. Also, second information computing section 220-2receives as input information X_(2,i) at point in time i and calculatesterm X₂(D) of equation 30-2. Also, third information computing section220-3 receives as input information X_(3,i) at point in time i andcalculates term X₃(D) of equation 30-3. Also, parity computing section230 receives as input parity P_(i-1) at point in time i−1 and calculatesterm P(D) of equations 30-1 to 30-3. Also, adder 240 finds, as parityP_(i) at point in time i, the exclusive OR of the computation results offirst information computing section 220-1, second information computingsection 220-2 and third information computing section 220-3 and thecomputation result of parity computing section 230.

With this configuration, upon creating an LDPC-CC supporting differentcoding rates, it is possible to share the configurations of informationcomputing sections according to the above explanation, so that it ispossible to provide an LDPC-CC encoder and decoder that can support aplurality of coding rates in a small computational complexity.

Also, in a case where A_(X1,k)(D) to A_(Xq-1,k)(D) are set to satisfythe above <Condition #1> to <Condition #6> described with the aboveLDPC-CCs of good characteristics, it is possible to provide an encoderand decoder that can support different coding rates in a smallcomputational complexity and provide data of good received quality inthe receiver. Here, as described in Embodiment 2, the method ofgenerating an LDPC-CC is not limited to the above case of LDPC-CCs ofgood characteristics

Also, by adding log likelihood ratio setting section 310 to the decoderconfiguration based on the maximum coding rate among coding rates forsharing decoder circuits, decoder 300 in FIG. 15 can perform decoding inaccordance with a plurality of coding rates. Also, according to a codingrate, log likelihood ratio setting section 310 sets log likelihoodratios for (q−2) items of information from information X_(r,i) toinformation X_(q-1,i) at point in time i, to predetermined values.

Also, although a case has been described above where the maximum codingrate supported by encoder 200 is ¾, the supported maximum coding rate isnot limited to this, and is equally applicable to a case where a codingrate of (q−1)/q (where q is an integer equal to or greater than 5) issupported (here, it naturally follows that it is possible to set themaximum coding rate to ⅔). In this case, essential requirements are thatencoder 200 employs a configuration including first to (q−1)-thinformation computing sections, and that adder 240 finds, as parity P₁at point in time i, the exclusive OR of the computation results of firstto (q−1)-th information computing sections and the computation result ofparty computing section 230.

Also, if all of coding rates supported by the transmitting and receivingapparatuses (encoder/decoder) are associated with codes based on themethods described with above Embodiment 2, by providing anencoder/decoder of the highest coding rate among the supported codingrates, it is possible to easily support coding and decoding in aplurality of coding rates and, at this time, provide an advantage ofreducing computational complexity significantly.

Also, although an example case has been described above where thedecoding scheme is sum-product decoding, the decoding method is notlimited to this, and it is equally possible to implement the presentinvention using decoding methods by a message-passing algorithm such asmin-sum decoding, normalized BP (Belief Propagation) decoding, shuffledBP decoding and offset BP decoding, as shown in Non-Patent Literature 5to Non-Patent Literature 7.

Next, a case will be explained where the present invention is applied toa communication apparatus that adaptively switches the coding rateaccording to the communication condition. Also, an example case will beexplained where the present invention is applied to a radiocommunication apparatus, the present invention is not limited to this,but is equally applicable to a PLC (Power Line Communication) apparatus,a visible light communication apparatus or an optical communicationapparatus.

FIG. 18 shows the configuration of communication apparatus 400 thatadaptively switches a coding rate. Coding rate determining section 410of communication apparatus 400 in FIG. 18 receives as input a receivedsignal transmitted from a communication apparatus of the communicatingparty (e.g. feedback information transmitted from the communicatingparty), and performs reception processing of the received signal.Further, coding rate determining section 410 acquires information of thecommunication condition with the communication apparatus of thecommunicating party, such as a bit error rate, packet error rate, frameerror rate and reception field intensity (from feedback information, forexample), and determines a coding rate and modulation scheme from theinformation of the communication condition with the communicationapparatus of the communicating party. Further, coding rate determiningsection 410 outputs the determined coding rate and modulation scheme toencoder 200 and modulating section 420 as a control signal.

Using, for example, the transmission format shown in FIG. 19, codingrate determining section 410 includes coding rate information in controlinformation symbols and reports the coding rate used in encoder 200 tothe communication apparatus of the communicating party. Here, as is notshown in FIG. 19, the communicating party includes, for example, knownsignals (such as a preamble, pilot symbol and reference symbol), whichare necessary in demodulation or channel estimation.

In this way, coding rate determining section 410 receives a modulationsignal transmitted from communication apparatus 500 of the communicatingparty, and, by determining the coding rate of a transmitted modulationsignal based on the communication condition, switches the coding rateadaptively. Encoder 200 performs LDPC-CC coding in the above steps,based on the coding rate designated by the control signal. Modulatingsection 420 modulates the encoded sequence using the modulation schemedesignated by the control signal.

FIG. 20 shows a configuration example of a communication apparatus ofthe communicating party that communicates with communication apparatus400. Control information generating section 530 of communicationapparatus 500 in FIG. 20 extracts control information from a controlinformation symbol included in a baseband signal. The controlinformation symbol includes coding rate information. Control informationgenerating section 530 outputs the extracted coding rate information tolog likelihood ratio generating section 520 and decoder 300 as a controlsignal.

Receiving section 510 acquires a baseband signal by applying processingsuch as frequency conversion and quadrature demodulation to a receivedsignal for a modulation signal transmitted from communication apparatus400, and outputs the baseband signal to log likelihood ratio generatingsection 520. Also, using known signals included in the baseband signal,receiving section 510 estimates channel variation in a channel (e.g.radio channel) between communication apparatus 400 and communicationapparatus 500, and outputs an estimated channel estimation signal to loglikelihood ratio generating section 520.

Also, using known signals included in the baseband signal, receivingsection 510 estimates channel variation in a channel (e.g. radiochannel) between communication apparatus 400 and communication apparatus500, and generates and outputs feedback information (such as channelvariation itself, which refers to channel state information, forexample) for deciding the channel condition. This feedback informationis transmitted to the communicating party (i.e. communication apparatus400) via a transmitting apparatus (not shown), as part of controlinformation. Log likelihood ratio generating section 520 calculates thelog likelihood ratio of each transmission sequence using the basebandsignal, and outputs the resulting log likelihood ratios to decoder 300.

As described above, according to the coding rate (s−1)/s designated by acontrol signal, decoder 300 sets the log likelihood ratios forinformation from information X_(s,i) to information X_(s-1,i), topredetermined values, and performs BP decoding using the LDPC-CC paritycheck matrix based on the maximum coding rate among coding rates toshare decoder circuits.

In this way, the coding rates of communication apparatus 400 andcommunication apparatus 500 of the communicating party to which thepresent invention is applied, are adaptively changed according to thecommunication condition.

Here, the method of changing the coding rate is not limited to theabove, and communication apparatus 500 of the communicating party caninclude coding rate determining section 410 and designate a desiredcoding rate. Also, communication apparatus 400 can estimate channelvariation from a modulation signal transmitted from communicationapparatus 500 and determine the coding rate. In this case, the abovefeedback information is not necessary. In this case, the above-describedfeedback information is unnecessary.

Embodiment 4

An LDPC-CC having high error correction capability has been described inEmbodiment 1. The present embodiment will provide supplementalexplanation of an LDPC-CC of a time varying period of 3 having higherror correction capability. In the case of an LDPC-CC of a time varyingperiod of 3, when a regular LDPC code is generated, it is possible tocreate a code having high error correction capability.

The parity check polynomial of the LDPC-CC of a time varying period of 3is presented again.

When coding rate is ½:[42](D ^(a1) +D ^(a2) +D ^(a3))X(D)+(D ^(b1) +D ^(b2) +D^(b3))P(D)=0  (Equation 42-1)(D ^(A1) +D ^(A2) +D ^(A3))X(D)+(D ^(B1) +D ^(B2) +D^(B3))P(D)=0  (Equation 42-2)(D ^(α1) +D ^(α2) +D ^(α3))X(D)+(D ^(β1) +D ^(β2) +D^(β3))P(D)=0  (Equation 42-3)

When coding rate is (n−1)/n:[43](D ^(a1,1) +D ^(a1,2) +D ^(a1,3))X ₁(D)+(D ^(a2,1) +D ^(a2,2) +D^(a2,3))X ₂(D)+ . . . +(D ^(an-1,1) +D ^(an-1,2) +D ^(an-1,3))X_(n-1)(D)+(D ^(b1) +D ^(b2) +D ^(b3))P(D)=0  (Equation 43-1)(D ^(A1,1) +D ^(A1,2) +D ^(A1,3))X ₁(D)+(D ^(A2,1) +D ^(A2,2) +D^(A2,3))X ₂(D)+ . . . +(D ^(An-1,1) +D ^(An-1,2) +D ^(An-1,3))X_(n-1)(D)+(D ^(B1) +D ^(B2) +D ^(B3))P(D)=0  (Equation 43-2)(D ^(α1,1) +D ^(α1,2) +D ^(α1,3))X ₁(D)+(D ^(α2,1) +D ^(α2,2) +D^(α2,3))X ₂(D)+ . . . +(D ^(αn-1,1) +D ^(αn-1,2) +D ^(αn-1,3))X_(n-1)(D)+(D ^(β1) +D ^(β2) +D ^(β3))P(D)=0  (Equation 43-3)

Here, to make sure that the parity check matrix becomes a full-rankmatrix and parity bits are sequentially easily obtained, it is assumedthat the following conditions hold true.

b3=0, that is, D^(b3)=1

B3=0, that is, D^(B3)=1

β3=0, that is, D^(β3)=1

Furthermore, to make the relationship between information and parityeasier to understand, the following conditions may preferably hold true.

ai, 3=0, that is, D^(ai,3)=1 (i=1, 2, . . . , n−1)

Ai, 3=0, that is, D^(Ai,3)=1 (i=1, 2, . . . , n−1)

αi, 3=0, that is, D^(αi,3)=1 (i=1, 2, . . . , n−1)

where, ai,3%3=0, Ai,3%3=0, and αi,3%3=0 may hold true.

At this time, to generate a regular LDPC code having high errorcorrection capability, the following conditions need to be satisfied byreducing the number of loops 6 in a Tanner graph.

That is, when attention is focused on the coefficient of information Xk(k=1, 2, . . . , n−1), one of #Xk1 to #Xk14 needs to be satisfied.

#Xk1: (ak,1%3, ak,2%3)=[0, 1], (Ak,1%3, Ak,2%3)=[0, 1], (αk,1%3,αk,2%3)=[0, 1]

#Xk2: (ak,1%3, ak,2%3)=[0, 1], (Ak,1%3, Ak,2%3)=[0, 2], (αk,1%3,αk,2%3)=[1, 2]

#Xk3: (ak,1%3, ak,2%3)=[0, 1], (Ak,1%3, Ak,2%3)=[1, 2], (αk,1%3,αk,2%3)=[1, 1]

#Xk4: (ak,1%3, ak,2%3)=[0, 2], (Ak,1%3, Ak,2%3)=[1, 2], (αk,1%3,αk,2%3)=[0, 1]

#Xk5: (ak,1%3, ak,2%3)=[0, 2], (Ak,1%3, Ak,2%3)=[0, 2], (αk,1%3,αk,2%3)=[0, 2]

#Xk6: (ak,1%3, ak,2%3)=[0, 2], (Ak,1%3, Ak,2%3)=[2, 2], (αk,1%3,αk,2%3)=[1, 2]

#Xk7: (ak,1%3, ak,2%3)=[1, 1], (Ak,1%3, Ak,2%3)=[0, 1], (αk,1%3,αk,2%3)=[1, 2]

#Xk8: (ak,1%3, ak,2%3)=[1, 1], (Ak,1%3, Ak,2%3)=[1, 1], (αk,1%3,αk,2%3)=[1, 1]

#Xk9: (ak,1%3, ak,2%3)=[1, 2], (Ak,1%3, Ak,2%3)=[0, 1], (αk,1%3,αk,2%3)=[0, 2]

#Xk10: (ak,1%3, ak,2%3)=[1, 2], (Ak,1%3, Ak,2%3)=[0, 2], (αk,1%3,αk,2%3)=[2, 2]

#Xk11: (ak,1%3, ak,2%3)=[1, 2], (Ak,1%3, Ak,2%3)=[1, 1], (αk,1%3,αk,2%3)=[0, 1]

#Xk12: (ak,1%3, ak,2%3)=[1, 2], (Ak,1%3, Ak,2%3)=[1, 2], (αk,1%3,αk,2%3)=[1, 2]

#Xk13: (ak,1%3, ak,2%3)=[2, 2], (Ak,1%3, Ak,2%3)=[1, 2], (αk,1%3,αk,2%3)=[0, 2]

#Xk14: (ak,1%3, ak,2%3)=[2, 2], (Ak,1%3, Ak,2%3)=[2, 2], (αk,1%3,αk,2%3)=[2, 2]

When a=b in the above description, (x, y)=[a, b] represents x=y=a(=b)and when a≠b, (x, y)=[a, b] represents x=a, y=b or x=b, y=a (the sameapplies hereinafter).

Similarly, when attention is focused on the coefficient of parity, oneof #P1 to #P14 needs to be satisfied.

#P1: (b1%3, b2%3)=[0, 1], (B1%3, B2%3)=[0, 1], (β1%3, β2%3)=[0, 1]

#P2: (b1%3, b2%3)=[0, 1], (B1%3, B2%3)=[0, 2], (β1%3, β2%3)=[1, 2]

#P3: (b1%3, b2%3)=[0, 1], (B1%3, B2%3)=[1, 2], (β1%3, β2%3)=[1, 1]

#P4: (b1%3, b2%3)=[0, 2], (B1%3, B2%3)=[1, 2], (β1%3, β2%3)=[0, 1]

#P5: (b1%3, b2%3)=[0, 2], (B1%3, B2%3)=[0, 2], (β1%3, β2%3)=[0, 2]

#P6: (b1%3, b2%3)=[0, 2], (B1%3, B2%3)=[2, 2], (β1%3, β2%3)=[1, 2]

#P7: (b1%3, b2%3)=[1, 1], (B1%3, B2V03)=[0, 1], (β1%3, β2%3)=[1, 2]

#P8: (b1%3, b2%3)=[1, 1], (B1%3, B2%3)=[1, 1], (β1%3, β2%3)=[1, 1]

#P9: (b1%3, b2%3)=[1, 2], (B1%3, B2%3)=[0, 1], (β1%3, β2%3)=[0, 2]

#P10: (b1%3, 62%3)=[1, 2], (B1%3, B2%3)=[0, 2], (β1%3, β2%3)=[2, 2]

#P11: (b1%3, b2%3)=[1, 2], (B1%3, B2%3)=[1, 1], (β1%3, β2%3)=[0, 1]

#P12: (b1%3, b2%3)=[1, 2], (B1%3, B2%3)=[1, 2], (β1%3, β2%3)=[1, 2]

#P13: (b1%3, b2%3)=[2, 2], (B1%3, B2%3)=[1, 2], (β1%3, β2%3)=[0, 2]

#P14: (b1%3, b2%3)=[2, 2], (B1%3, B2%3)=[2, 2], (β1%3, β2%3)=[2, 2]

The LDPC-CC with good characteristics described in Embodiment 1 is theLDPC-CC that satisfies the conditions of #Xk 12 and #P12 among the aboveconditions. Furthermore, when used together with Embodiment 2, thepresent embodiment can reduce the circuit scale of the encoder anddecoder when supporting a plurality of coding rates and obtain higherror correction capability.

The following is an example of parity check polynomial of an LDPC-CC ofa time varying period of 3 that satisfies the conditions of one of #Xk1to #Xk14 and one of #P1 to #P14.

Coding rate R=½:[44]A _(X1,1)(D)X ₁(D)+B ₁(D)P(D)=(D ²⁸⁶ +D ¹⁶⁴+1)X ₁(D)+(D ⁹² +D⁷+1)P(D)=0  (Equation 44-1)A _(X1,2)(D)X ₁(D)+B ₂(D)P(D)=(D ³⁷⁰ +D ³¹⁷+1)X ₁(D)+(D ⁹⁵ +D²²+1)P(D)=0  (Equation 44-2)A _(X1,3)(D)X ₁(D)+B ₃(D)P(D)=(D ³⁴⁶ +D ⁸⁶+1)X ₁(D)+(D ⁸⁸ +D²⁶+1)P(D)=0  (Equation 44-3)

Coding rate R=⅔:[45]A _(X1,1)(D)X ₁(D)+A _(X2,1)(D)X ₂(D)+B ₁(D)P(D)=(D ²⁸⁶ +D ¹⁶⁴+1)X₁(D)+(D ³⁸⁵ +D ²⁴²+1)X ₂(D)+(D ⁹² +D ⁷+1)P(D)=0  (Equation 45-1)A _(X1,2)(D)X ₁(D)+A _(X2,2)(D)X ₂(D)+B ₂(D)P(D)=(D ³⁷⁰ +D ³¹⁷+1)X₁(D)+(D ¹²⁵ +D ¹⁰³+1)X ₂(D)+(D ⁹⁵ +D ²²+1)P(D)=0  (Equation 45-2)A _(X1,3)(D)X ₁(D)+A _(X2,3)(D)X ₂(D)+B ₃(D)P(D)=(D ³⁴⁶ +D ⁸⁶+1)X₁(D)+(D ³¹⁹ +D ²⁹⁰+1)X ₂(D)+(D ⁸⁸ +D ²⁶+1)P(D)=0  (Equation 45-3)

Coding rate R=¾:[46]A _(X1,1)(D)X ₁(D)+A _(X2,1)(D)X ₂(D)+A _(X3,1)(D)X ₃(D)B ₁(D)P(D)=(D²⁸⁶ +D ¹⁶⁴+1)X ₁(D)+(D ³⁸⁵ +D ²⁴²+1)X ₂(D)+(D ³⁴³ +D ²⁸⁴+1)X ₃(D ⁹² +D⁷+1)P(D)=0  (Equation 46-1)A _(X1,2)(D)X ₁(D)+A _(X2,2)(D)X ₂(D)+A _(X3,2)(D)X ₃(D)B ₂(D)P(D)=(D³⁷⁰ +D ³¹⁷+1)X ₁(D)+(D ¹²⁵ +D ¹⁰³+1)X ₂(D)+(D ²⁵⁹ +D ¹⁴+1)X ₃(D)+(D ⁹⁵+D ²²+1)P(D)=0  (Equation 46-2)A _(X1,3)(D)X ₁(D)+A _(X2,3)(D)X ₂(D)+A _(X3,3)(D)X ₃(D)B ₃(D)P(D)=(D³⁴⁶ +D ⁸⁶+1)X ₁(D)+(D ³¹⁹ +D ²⁹⁰+1)X ₂(D)+(D ¹⁴⁵ +D ¹¹+1)X ₃(D)+(D ⁸⁸ +D²⁶+1)P(D)=0  (Equation 46-3)

Coding rate R=⅘:[47]A _(X1,1)(D)X ₁(D)+A _(X2,1)(D)X ₂(D)+A _(X3,1)(D)X ₃(D)+A _(X4,1)(D)X₄(D)+B ₁(D)P(D)=(D ²⁸⁶ +D ¹⁶⁴+1)X ₁(D)+(D ³⁸⁵ +D ²⁴²+1)X ₂(D)+(D ³⁴³ +D²⁸⁴+1)X ₃(D)+(D ²⁸³ +D ⁶⁸+1)X ₄(D)+(D ⁹² +D ⁷+1)P(D)=0  (Equation 47-1)A _(X1,2)(D)X ₁(D)+A _(X2,2)(D)X ₂(D)+A _(X3,2)(D)X ₃(D)+A _(X4,2)(D)X₄(D)+B ₂(D)P(D)=(D ³⁷⁰ +D ³¹⁷+1)X ₁(D)+(D ¹²⁵ +D ¹⁰³+1)X ₂(D)+(D ²⁵⁹ +D¹⁴+1)X ₃(D)+(D ²⁵⁶ +D ¹⁸⁸+1)X ₄(D)+(D ⁹⁵ +D ²²+1)P(D)=0  (Equation 47-2)A _(X1,3)(D)X ₁(D)+A _(X2,3)(D)X ₂(D)+A _(X3,3)(D)X ₃(D)+A _(X4,3)(D)X₄(D)+B ₃(D)P(D)=(D ³⁴⁶ +D ⁸⁶+1)X ₁(D)+(D ³¹⁹ +D ²⁹⁰+1)X ₂(D)+(D ¹⁴⁵ +D¹¹+1)X ₃(D)+(D ²⁸⁷ +D ⁷³+1)X ₄(D)+(D ⁸⁸ +D ²⁶+1)P(D)=0  (Equation 47-3)

Since the parity check polynomial of the above LDPC-CC satisfies theconditions described in Embodiment 2, it is possible to realize thesharing of encoder and decoder circuits.

When the parity check polynomials of the LDPC-CC shown in equation 44-i,equation 45-i, equation 46-i and equation 47-i (i=1, 2, 3) are used, ithas been confirmed that the termination number required varies dependingon the number of bits of data (information) X (hereinafter referred toas “information size”) as shown in FIG. 21. Here, the termination numberrefers to the number of parity bits generated by virtual knowninformation bit “0” after performing the above-describedinformation-zero-termination and is the number of redundant bitsactually transmitted. In FIG. 21, Real R (effective coding rate)represents a coding rate when the termination sequence made up ofredundant bits is taken into consideration.

The following is another example of parity check polynomial of anLDPC-CC of a time varying period of 3 that satisfies the conditions ofone of #Xk1 to #Xk14 and one of #P1 to #P14.

Coding rate R=½:[48]A _(X1,1)(D)X ₁(D)+B ₁(D)P(D)=(D ²¹⁴ +D ¹⁸⁵+1)X ₁(D)+(D ²¹⁵ +D¹⁴⁵+1)P(D)=0  (Equation 48-1)A _(X1,2)(D)X ₁(D)+B ₂(D)P(D)=(D ¹⁶⁰ +D ⁶²+1)X ₁(D)+(D ²⁰⁵ +D¹²⁷+1)P(D)=0  (Equation 48-2)A _(X1,3)(D)X ₁(D)+B ₃(D)P(D)=(D ¹⁹⁶ +D ¹⁴³+1)X ₃(D)+(D ²¹⁸ +D¹¹⁹+1)P(D)=0  (Equation 48-3)

Coding rate R=⅔:[49]A _(X1,1)(D)X ₁(D)+A _(X2,1)(D)X ₂(D)+B ₁(D)P(D)=(D ²¹⁴ +D ¹⁸⁵+1)X₁(D)+(D ¹⁹⁴ +D ⁶⁷+1)X ₂(D)+(D ²¹⁵ +D ¹⁴⁵+1)P(D)=0  (Equation 49-1)A _(X1,2)(D)X ₁(D)+A _(X2,2)(D)X ₂(D)+B ₂(D)P(D)=(D ¹⁶⁰ +D ⁶²+1)X₁(D)+(D ²²⁶ +D ²⁰⁹+1)X ₂(D)+(D ²⁰⁶ +D ¹²⁷+1)P(D)=0  (Equation 49-2)A _(X1,3)(D)X ₁(D)+A _(X2,3)(D)X ₂(D)+B ₃(D)P(D)=(D ¹⁹⁶ +D ¹⁴³+1)X₁(D)+(D ¹¹⁵ +D ¹⁰⁴+1)X ₂(D)+(D ²¹¹ +D ¹¹⁹+1)P(D)=0  (Equation 49-3)

Coding rate R=¾:[50]A _(X1,1)(D)X ₁(D)+A _(X2,1)(D)X ₂(D)+A _(X3,1)(D)X ₃(D)+B ₁(D)P(D)=(D²¹⁴ +D ¹⁸⁵+1)X ₁(D)+(D ¹⁹⁴ +D ⁶⁷+1)X ₂(D)+(D ¹⁹⁶ +D ⁶⁸+1)X ₃(D)+(D ²¹⁵+D ¹⁴⁵+1)P(D)=0  (Equation 50-1)A _(X1,2)(D)X ₁(D)+A _(X2,2)(D)X ₂(D)+A _(X3,2)(D)X ₃(D)+B ₂(D)P(D)=(D¹⁶⁰ +D ⁶²+1)X ₁(D)+(D ²²⁶ +D ²⁰⁹+1)X ₂(D)+(D ⁹⁸ +D ³⁷+1)X ₃(D)+(D ²⁰⁶ +D¹²⁷+1)P(D)=0  (Equation 50-2)A _(X1,3)(D)X ₁(D)+A _(X2,3)(D)X ₂(D)+A _(X3,3)(D)X ₃(D)+B ₃(D)P(D)=(D¹⁹⁶ +D ¹⁴³+1)X ₁(D)+(D ¹¹⁵ +D ¹⁰⁴+1)X ₂(D)+(D ¹⁷⁶ +D ¹³⁶+1)X ₃(D)+(D ²¹¹+D ¹¹⁹+1)P(D)=0  (Equation 50-3)

Coding rate R=⅘:[51]A _(X1,1)(D)X ₁(D)+A _(X2,1)(D)X ₂(D)+A _(X3,1)(D)X ₃(D)+A _(X4,1)(D)X₄(D)+B ₁(D)P(D)=(D ²¹⁴ +D ¹⁸⁵+1)X ₁(D)+(D ¹⁹⁴ +D ⁶⁷+1)X ₂(D)+(D ¹⁹⁶ +D⁶⁸+1)X ₃(D)+(D ²¹⁷ +D ¹²²+1)X ₄(D)+(D ²¹⁵ +D ¹⁴⁵+1)P(D)=0  (Equation51-1)A _(X1,2)(D)X ₁(D)+A _(X2,2)(D)X ₂(D)+A _(X3,2)(D)X ₃(D)+A _(X4,2)(D)X₄(D)+B ₂(D)P(D)=(D ¹⁶⁰ +D ⁶²+1)X ₁(D)+(D ²²⁶ +D ²⁰⁹+1)X ₂(D)+(D ⁹⁸ +D³⁷+1)X ₃(D)+(D ⁷¹ +D ³⁴+1)X ₄(D)+(D ²⁰⁶ +D ¹²⁷+1)P(D)=0  (Equation 51-2)A _(X1,3)(D)X ₁(D)+A _(X2,3)(D)X ₂(D)+A _(X3,3)(D)X ₃(D)+A _(X4,3)(D)X₄(D)+B ₃(D)P(D)=(D ¹⁹⁶ +D ¹⁴³+1)X ₁(D)+(D ¹¹⁵ +D ¹⁰⁴+1)X ₂(D)+(D ¹⁷⁶ +D¹³⁶+1)X ₃(D)+(D ²¹² +D ¹⁸⁷+1)X ₄(D)+(D ³¹¹ +D ¹¹⁹+1)P(D)=0  (Equation51-3)

FIG. 22 shows an example of termination number necessary to use paritycheck polynomials of the LDPC-CC shown in equation 48-i, equation 49-i,equation 50-i and equation 51-i (i=1, 2, 3).

FIG. 23 shows the relationship between information size I_(s) andtermination number m_(t) shown in equation 48-i, equation 49-i, equation50-i and equation 51-i (i=1, 2, 3) at each coding rate. Assuming thenumber of virtual known information bits (“0”) inserted to create atermination sequence is m_(z), the following relationship holds truebetween m_(t) and m_(z) when the coding rate is (n−1)/n.[52]m _(z)=(n−1)m _(t) (k=0)  (Equation 52-1)m _(z)=(n−1)m _(t)+(n−1)−k (k≠0)  (Equation 52-2)

where k=I_(s)%(n−1).

Embodiment 5

The present embodiment will describe a communication apparatus andcommunication method when using the LDPC-CC with good characteristicsdescribed in Embodiment 4, which can prevent the error correctioncapability from deteriorating and prevent information transmissionefficiency from deteriorating.

It has been confirmed from FIG. 21 and FIG. 22 that the terminationnumber necessary to perform information-zero-termination variesdepending on the information size. Therefore, to uniformly fix thetermination number irrespective of the information size and preventerror correction capability from deteriorating, the termination numbermay have to be set to a large value, and therefore Real R (effectivecoding rate) may deteriorate and information transmission efficiency maydeteriorate.

Thus, the present embodiment will describe a communication apparatus andcommunication method that change the termination number transmitted asredundant bits according to the information size. It is thereby possibleto prevent error correction capability from deteriorating and preventinformation transmission efficiency from deteriorating.

FIG. 24 is a block diagram showing the main configuration ofcommunication apparatus 600 according to the present embodiment.

Coding rate setting section 610 receives a control information signalincluding information of a coding rate set by communication apparatus600 or a feedback signal transmitted from the communication apparatuswhich is the communicating party as input. When the control informationsignal is inputted, coding rate setting section 610 sets a coding ratefrom the information of coding rates included in the control informationsignal.

Upon receiving a feedback signal, coding rate setting section 610acquires information of the communication situation between thecommunication apparatus and the communication apparatus which is thecommunicating party included in the feedback signal, for example,information that allows communication quality such as bit error rate,packet error rate, frame error rate, reception electric field strengthto be estimated, and sets the coding rate based on the information ofthe communication situation between the communication apparatus and thecommunication apparatus which is the communicating party. Coding ratesetting section 610 includes the information of the set coding rate inthe set coding rate signal and outputs the set coding rate signal totermination sequence length determining section 631 and parity computingsection 632 in encoder 630. Furthermore, coding rate setting section 610outputs the information of the set coding rate to transmissioninformation generation and information length detection section 620.

Transmission information generation and information length detectionsection 620 generates or acquires transmission data (information) andoutputs an information sequence made up of transmission data(information) to parity computing section 632. Furthermore, transmissioninformation generation and information length detection section 620detects the sequence length of transmission data (information)(hereinafter referred to as “information length”), that is, informationsize, includes the information of the detected information size in theinformation length signal and outputs an information length signal totermination sequence length determining section 631. Furthermore,transmission information generation and information length detectionsection 620 adds a known information sequence made up of knowninformation bits (e.g. “0”) necessary to generate redundant bitscorresponding to the termination sequence length reported fromtermination sequence length determining section 631 at the rearmost endof the information sequence.

Termination sequence length determining section 631 determines thetermination sequence length (termination number) according to theinformation size indicated by an information length signal and thecoding rate indicated by the set coding rate signal. A specific methodof determining the termination sequence length will be described later.Termination sequence length determining section 631 includes thedetermined termination sequence length in the termination sequencelength signal and outputs the termination sequence length signal totransmission information generation and information length detectionsection 620 and parity computing section 632.

Parity computing section 632 calculates parity corresponding to theinformation sequence and known information sequence and outputs theparity obtained to modulation section 640.

Modulation section 640 applies modulation processing to the informationsequence and parity (including the termination sequence).

Although there is a description “information length signal” in FIG. 24,the signal is not limited to this, but any signal may be adopted if itis information that serves as an index to control the terminationsequence length. For example, it may be possible to calculate a framelength of the transmission signal from information (length information)on the sum of the number of pieces of information except termination andparity, the number of pieces of information and modulation scheme, anddesignate the frame length a substitute for the information lengthsignal.

Next, a method of determining the termination sequence length bytermination sequence length determining section 631 will be describedusing FIG. 25. FIG. 25 shows an example case where the terminationsequence length is switched in two stages based on the information sizeand each coding rate. FIG. 25 presupposes that the minimum informationsize of communication apparatus 600 is set to 512 bits. However, theminimum size need not always be set.

In FIG. 25, α is the information length of transmission data(information) that should be transmitted. For example, when the codingrate is ½, termination sequence length determining section 631 sets thetermination sequence length to 380 bits when 512≦α≦0.1023 andtermination sequence length determining section 631 sets the terminationsequence length to 340 hits when 1024<α. When termination sequencelength determining section 631 sets the termination sequence lengthbased on information length α of transmission data (information) in thisway, the termination sequence length can be set to a sequence lengththat does not cause deterioration of error correction capability and canprevent information transmission efficiency from deteriorating.

An example case has been described above where the termination sequencelength is switched in two stages at each coding rate, but the presentinvention is not limited to this, and the termination sequence lengthmay be switched in three stage as shown in FIG. 26 or more stages. Thus,by switching the termination sequence length (termination number) in aplurality of stages based on the information length (information size),it is possible to set the termination sequence length to a sequencelength that does not cause deterioration of error correction capabilityand can prevent information transmission efficiency from deteriorating.

Communication apparatus 600 includes the information of the coding ratein a symbol relating to the coding rate using a transmission format asshown, for example, in FIG. 27 and thereby reports the coding rate usedby encoder 630 to the communication apparatus which is the communicatingparty. Furthermore, communication apparatus 600 includes information ofthe information length (information size) in a symbol relating to theinformation size and thereby reports the information of the informationlength (information size) to the communication apparatus which is thecommunicating party. Furthermore, communication apparatus 600 includesthe modulation scheme, transmission method or information foridentifying the communicating party in the control information symbolsand reports the symbols to the communication apparatus which is thecommunicating party. Furthermore, communication apparatus 600 includesthe information sequence and parity in data symbols and reports the datasymbols to the communication apparatus which is the communicating party.

FIG. 28 shows a configuration example of communication apparatus 700which is the communicating party communicating with communicationapparatus 600. Components of communication apparatus 700 in FIG. 28which are common to those of FIG. 20 are assigned the same referencenumerals as those in FIG. 20 and descriptions thereof will be omitted.Communication apparatus 700 in FIG. 28 is provided with controlinformation generating section 710 and decoder 720 instead of controlinformation generating section 530 and decoder 300 of communicationapparatus 500 in FIG. 20.

Control information generating section 710 extracts information of thecoding rate from a symbol relating the coding rate obtained bydemodulating (and decoding) a baseband signal. Furthermore, controlinformation generating section 710 extracts information of theinformation length (information size) from a symbol relating to theinformation size obtained by demodulating (and decoding) the basebandsignal. Furthermore, control information generating section 710 extractsthe modulation scheme, transmission method or information foridentifying the communicating party from the control informationsymbols. Control information generating section 710 outputs a controlsignal including the information of the extracted coding rate and theinformation of the information length (information size) to loglikelihood ratio generating section 520 and decoder 720.

Decoder 720 stores a table showing the relationship between theinformation size and termination sequence length at each coding rateshown in FIG. 25 or FIG. 26, and determines the termination sequencelength included in data symbols from this table, information of thecoding rate and information of the information length (informationsize). Decoder 720 performs BP decoding based on the coding rate and thedetermined termination sequence length. This allows communicationapparatus 700 to perform decoding with high error correction capability.

FIG. 29 and FIG. 30 are diagrams showing an example of information flowbetween communication apparatus 600 and communication apparatus 700.FIG. 29 is different from FIG. 30 as to which of communication apparatus600 and communication apparatus 700 sets the coding rate. To be morespecific, FIG. 29 shows the information flow when communicationapparatus 600 determines the coding rate and FIG. 30 shows theinformation flow when communication apparatus 700 determines the codingrate.

As described so far, in the present embodiment, termination sequencelength determining section 631 determines the sequence length of atermination sequence transmitted by being added at the rear end of aninformation sequence according to the information length (informationsize) and coding rate and parity computing section 632 applies LDPC-CCcoding to the information sequence and a known information sequencenecessary to generate a termination sequence corresponding to thedetermined termination sequence length and computes a parity sequence.This makes it possible to prevent deterioration of error correctioncapability and prevent deterioration of information transmissionefficiency.

Embodiment 6

A case has been described in Embodiment 5 where a termination sequencelength added at the rear end of an information sequence is determined(changed) according to the information length (information size) andcoding rate. This makes it possible to prevent deterioration of errorcorrection capability and avoid deterioration of informationtransmission efficiency.

The present embodiment will describe a case where a limit is set toavailable coding rates when the termination sequence length is changedaccording to the information length (information size) as in the case ofEmbodiment 5. This makes it possible to avoid deterioration of errorcorrection capability.

As in the case of FIG. 21, FIG. 31 shows the relationship between thetermination number and coding rate necessary to use parity checkpolynomials of the LDPC-CCs shown in equation 44-i, equation 45-i,equation 46-i and equation 47-i (i=1, 2, 3). As is clear from FIG. 31,when the information size is 512 bits, 1024 bits or 2048 bits, if theeffective coding rate (Real R) of a coding rate of ¾ is compared to theeffective coding rate of a coding rate of ⅘, there is no significantdifference between the two. When, for example, the information size is1024 bits, the effective coding rate is 0.5735 for the coding rate of ¾,while the effective coding rate is 0.5626 for the coding rate of ⅘, thedifference being as small as 0.01. Furthermore, the effective codingrate of coding rate ¾ is greater than the effective coding rate ofcoding rate ⅘, that is, the magnitude of the effective coding rate isinverted. Therefore, depending on the information size, use of codingrate ¾ may not be suited to obtaining high error correction capabilityor improving transmission efficiency.

FIG. 32A, FIG. 32B, FIG. 32C and FIG. 32D show bit error rate(BER)/block error rate (BLER) characteristics when the terminationsequences of the sequence lengths shown in FIG. 31 are added toinformation sequences having information sizes of 512 bits, 1024 bits,2048 bits and 4096 bits. In FIG. 32A, FIG. 32B, FIG. 32C and FIG. 32D,the horizontal axis shows an SNR (Signal-to-Noise power ratio) [dB] andthe vertical axis shows BER/BLER characteristics, and solid lines showbit error rate characteristics and broken lines show block error ratecharacteristics. Furthermore, in FIG. 32A, FIG. 32B, FIG. 32C and FIG.32D, TMN represents a termination number.

As is clear from FIG. 32A, FIG. 32B, FIG. 32C and FIG. 32D, when thetermination sequence is taken into consideration, the BER/BLERcharacteristics of coding rate R=¾ are better than the BER/BLERcharacteristics of coding rate R=⅘ no matter what the information sizemay be.

From these two aspects, in order to realize improvement of errorcorrection capability and improvement of information transmissionefficiency simultaneously, coding rate R=⅘ is not supported when theinformation size is less than 4096 bits, that is, only coding rates R=½,⅔ and ¾ are supported when the information size is less than 4096 bitsand coding rates R=½, ⅔, ¾ and ⅘ are supported when the information sizeis equal to or greater than 4096 bits, and for this reason, coding rateR=⅘ having poorer transmission efficiency than coding rate R=¾ is nolonger used when the information size is less than 4096 bits, and it isthereby possible to realize improvement of error correction capabilityand improvement of information transmission efficiency simultaneously.

Furthermore, it is clear from FIG. 32A, FIG. 32B, FIG. 32C and FIG. 32Dthat the BER/BLER characteristics (see FIG. 32A) when the informationsize is 512 bits are notably better than the BER/BLER characteristics inother information sizes. When, for example, the information size is 512bits, the BER characteristics of coding rate ⅔ are substantially thesame as the BER/BLER characteristics of coding rate ½ when theinformation size is 1024 bits, and when the information size is 512bits, the characteristics actually may not have to be as better as theBER/BLER characteristics of coding rate ½. Since the propagationefficiency decreases as the coding rate decreases, when, for example,the information size is 512 bits, a method of not supporting coding rate½ may also be adopted taking these points into account.

FIG. 33 is a table of correspondence between information sizes andsupported coding rates. As shown in FIG. 33, some coding rates are notsupported for certain information sizes. If supported coding rates areconstant irrespective of the information size, communication apparatus600 and communication apparatus 700 can communicate with each other inboth cases of FIG. 29 and FIG. 30. However, as shown in FIG. 33, sincesome coding rates are not supported for certain information sizes in thepresent embodiment, the designated coding rates need to be adjusted. Thecommunication apparatus according to the present embodiment will bedescribed below.

FIG. 34 is a block diagram showing the main configuration ofcommunication apparatus 600A according to the present embodiment. Incommunication apparatus 600A in FIG. 34, components common to those inFIG. 24 are assigned the same reference numerals as those in FIG. 24 anddescriptions thereof will be omitted. Communication apparatus 600A inFIG. 34 is provided with encoder 630A instead of encoder 630 in FIG. 24.Encoder 630A adopts a configuration with coding rate adjusting section633 added to encoder 630.

Coding rate adjusting section 633 adjusts a coding rate included in aset coding rate signal inputted from coding rate setting section 610based on the information length (information size) included in theinformation length signal inputted from transmission informationgeneration and information length detection section 620. To be morespecific, coding rate adjusting section 633 stores the table ofcorrespondence between information sizes and supported coding ratesshown in FIG. 33 and adjusts the coding rate by checking the coding rateset based on a control information signal or feedback signal against thecorrespondence table. When, for example, the information length(information size) is 1024 bits and the set coding rate signal indicatesa coding rate of ⅘, since a coding rate of ⅘ is not supported in thecorrespondence table, coding rate adjusting section 633 sets ¾ which isthe largest value among coding rates smaller than coding rate ⅘ as thecoding rate. As shown in FIG. 31, when the information length(information size) is 1024 bits, Real R when the coding rate is ⅘ is0.5626, which is smaller than Real R (0.5735) of coding rate ¾, and asshown in FIG. 32B, coding rate ¾ also has better BER/BLERcharacteristics. Therefore, when the information length (informationsize) is 1024, using coding rate ¾ instead coding rate ⅘ makes itpossible to prevent error correction capability from deteriorating andprevent information transmission efficiency from deteriorating.

In other words, when first coding rate (¾)<second coding rate (⅘), if afirst effective coding rate (0.5735) corresponding to the first codingrate (¾) is equivalent to a second effective coding rate (0.5626)corresponding to the second coding rate (⅘) and if the second codingrate is designated, coding rate adjusting section 633 adjusts the codingrate to the first coding rate. This makes it possible to prevent errorcorrection capability from deteriorating and prevent informationtransmission efficiency from deteriorating.

Furthermore, when, for example, the information length (informationsize) is 512 bits and the set coding rate signal indicates coding rate½, since coding rate ½ is not supported in the correspondence table,coding rate adjusting section 633 sets ⅔ which is the smallest valueamong coding rates greater than coding rate ½ as the coding rate. Asshown in FIG. 32A, since the BER/BLER characteristics at coding rate ½are extremely good, setting the coding rate to ⅔ also makes it possibleto prevent error correction capability from deteriorating and preventinformation transmission efficiency from deteriorating.

In other words, when the first coding rate having extremely goodBER/BLER characteristics is designated, coding rate adjusting section633 adjusts the coding rate to the second coding rate which is greaterthan the first coding rate and can secure predetermined channel quality.

Thus, the present embodiment is designed to change the number of codingrates supported by communication apparatus 600A based on the informationlength (information size). For example, in the example shown in FIG. 33,communication apparatus 600A only supports two coding rates when theinformation length (information size) is less than 512 bits, supportsthree coding rates when the information length (information size) isequal to or greater than 512 bits and less than 4096 bits, and supportsfour coding rates when the information length (information size) isequal to or greater than 4096. By changing supported coding rates, it ispossible to realize improvement of error correction capability andimprovement of information transmission efficiency simultaneously.

As shown above, according to the present embodiment, coding rateadjusting section 633 changes the number of coding rates supported bycommunication apparatus 600A according to the information length(information size) and adjusts the coding rate to one of the supportedcoding rates. This makes it possible to prevent error correctioncapability from deteriorating and prevent information transmissionefficiency from deteriorating.

Furthermore, communication apparatus 600A is designed to support codingrates of smaller values among coding rates of the same level ofeffective coding rates. Furthermore, communication apparatus 600A isalso designed not to include coding rates of extremely good BER/BLERcharacteristics in the supported coding rates but to support only codingrates that allow predetermined channel quality to be secured. This makesit possible to secure predetermined channel quality and preventdeterioration of transmission efficiency at the same time.

As described above, by changing the number of coding rates supportedaccording to the information length (information size), it is possibleto realize improvement of error correction capability and improvement ofinformation transmission efficiency simultaneously.

When the number of coding rates supported is changed according to theinformation length (information size), as shown in FIG. 29, ifcommunication apparatus 600A adjusts the coding rate, sets thetermination sequence length and transmits information of these codingrates and information of the information length (information size) (orinformation of termination sequence length) to communication apparatus700 which is the communicating party simultaneously, communicationapparatus 700 can perform decoding correctly.

The present embodiment may naturally be used together with Embodiment 5.That is, the termination number may be changed according to the codingrate and information size.

On the other hand, as shown in FIG. 30, if the communication apparatuswhich is the communicating party of communication apparatus 600 sets acoding rate before communication apparatus 600A determines aninformation length (information size) or if communication apparatus 600Asets a coding rate before communication apparatus 600A determines aninformation length (information size) as shown in FIG. 35, thecommunication apparatus which is the communicating party ofcommunication apparatus 600A needs to adjust the coding rate based onthe information length (information size). FIG. 36 is a block diagramshowing the configuration of communication apparatus 700A in this case.

In communication apparatus 700A in FIG. 36, components common to thosein FIG. 28 are assigned the same reference numerals and descriptionsthereof will be omitted. Communication apparatus 700A in FIG. 36 adoptsa configuration with coding rate adjusting section 730 added tocommunication apparatus 700 in FIG. 28.

A case will be described hereinafter where communication apparatus 600Asupports coding rates ½, ⅔ and ¾ when the information length(information size) is less than 4096 bits and supports coding rates ½,⅔, ¾ and ⅘ when the information length (information size) is 4097 bits.

At this time, it is assumed that the coding rate of an informationsequence to transmit is determined to be ⅘ before the information length(information size) is determined and communication apparatus 600A andcommunication apparatus 700A share the information of this coding rate.When the information length (information size) is 512 bits, coding rateadjusting section 633 of communication apparatus 600A adjusts the codingrate to ¾ as described above. If this rule is determined beforehandbetween communication apparatus 600A and communication apparatus 700A,communication apparatus 600A and communication apparatus 700A cancommunicate with each other correctly.

To be more specific, as in the case of coding rate adjusting section633, coding rate adjusting section 730 receives a control signalincluding the information of the coding rate and information of theinformation length (information size) as input and adjusts the codingrate based on the information length (information size). For example,when the information length (information size) is 512 bits and thecoding rate is ⅘, coding rate adjusting section 730 adjusts the codingrate to ¾. This makes it possible to prevent error correction capabilityfrom deteriorating and prevent information transmission efficiency fromdeteriorating.

As another coding rate adjusting method, a method of fixing thetermination number irrespective of coding rates can be considered. Inthe example in FIG. 21, when the information length (information size)is equal to or greater than 6144, the termination number is uniformly340 bits. Therefore, when the information length (information size) isequal to or greater than 6144 bits, coding rate adjusting section 633and coding rate adjusting section 730 may be adapted so as to fix thetermination number irrespective of coding rates. Furthermore, when theinformation length (information size) is less than 6144, coding rateadjusting section 633 and coding rate adjusting section 730 may also beadapted so as to support each coding rate using another parity checkpolynomial to which a termination number of 340 bits is fitted.Alternatively, a completely different code may be used. For example,block codes may be used.

Embodiment 7

The above embodiments have described an LDPC-CC for which circuitssupporting a plurality of coding rates equal to or greater than ½ can beshared between an encoder and a decoder. To be more specific, an LDPC-CCsupporting a coding rate of (n−1)/n (n=2, 3, 4, 5) for which circuitscan be shared has been described. The present embodiment will describe amethod of supporting a coding rate of ⅓.

FIG. 37 is a block diagram showing an example of the configuration of anencoder according to the present embodiment. In encoder 800 in FIG. 37,coding rate setting section 810 outputs a coding rate to control section820, parity computing section 830 and parity computing section 840.

When coding rate setting section 810 designates coding rates ½, ⅔, ¾ and⅘, control section 820 performs control so that information is notinputted to parity computing section 840. Furthermore, when coding rate⅓ is set, control section 820 performs control so that the sameinformation as the information inputted to parity computing section 830is inputted to parity computing section 840.

Parity computing section 830 is an encoder that obtains parity of codingrates ½, ⅔, ¾ and ⅘ defined by equation 44-i, equation 45-i, equation46-i and equation 47-i (i=1, 2, 3).

When coding rate setting section 810 designates coding rates ½, ⅔, ¾ and⅘, parity computing section 830 performs encoding based on thecorresponding parity check polynomials and outputs parity.

When coding rate setting section 810 designates coding rate ⅓, paritycomputing section 830 performs encoding based on the parity checkpolynomials of an LDPC-CC of a coding rate of ½ (defined by equation44-1, equation 44-2, equation 44-3) and of a time varying period of 3and outputs parity P.

Parity computing section 840 is an encoder that obtains parity of acoding rate of ½. When coding rate setting section 810 designates codingrates ½, ⅔, ¾ and ⅘, parity computing section 840 does not outputparity.

When coding rate setting section 810 designates coding rate ⅓, paritycomputing section 840 receives the same information as the informationinputted to parity computing section 830 as input, performs encodingbased on the parity check polynomials of an LDPC-CC of a coding rate of½ and a time varying period of 3 and outputs parity Pa.

Thus, since encoder 800 outputs information, parity P and parity Pa,encoder 800 can support coding rate ⅓.

FIG. 38 is a block diagram showing an example of the configuration of adecoder according to the present embodiment. Decoder 900 in FIG. 38 is adecoder corresponding to encoder 800 in FIG. 37.

Control section 910 receives coding rate information indicating a codingrate and a log likelihood ratio as input and performs control so thatwhen the coding rate is ½, ⅔, ¾ or ⅘, the log likelihood ratio is notinputted to BP decoding section 930. Furthermore, when the coding rateis ⅓, control section 910 performs control so that the same loglikelihood ratio as the log likelihood ratio inputted to BP decodingsection 920 is inputted to BP decoding section 930.

BP decoding section 920 operates at all coding rates. To be morespecific, when the coding rate is ⅓, BP decoding section 920 performs BPdecoding using the parity check polynomial of a coding rate of ½ used inparity computing section 830. Furthermore, when the coding rate is ⅓, BPdecoding section 920 outputs a log likelihood ratio corresponding toeach bit obtained by BP decoding to BP decoding section 930. On theother hand, when the coding rate is ½, ⅔, ¾ or ⅘, BP decoding section920 performs BP decoding using the parity check polynomial of codingrate ½, ⅔, ¾ or ⅘ used in parity computing section 830. BP decodingsection 920 performs iterative decoding a predetermined number of timesand then outputs the log likelihood ratio obtained.

BP decoding section 930 operates only when the coding rate is ⅓. To bemore specific, BP decoding section 930 performs BP decoding using theparity check polynomial of a coding rate of ½ used in parity computingsection 840, outputs a log likelihood ratio corresponding to each bitobtained by performing BP decoding to BP decoding section 920, performsiterative decoding a predetermined number of times and then outputs alog likelihood ratio obtained.

Thus, decoder 900 performs iterative decoding while switching betweenlog likelihood ratios, performs decoding such as turbo decoding andperforms decoding at a coding rate of ⅓.

Embodiment 8

Embodiment 2 has described an encoder that creates an LDPC-CC of a timevarying period of g (g is a natural number) supporting a plurality ofcoding rates of (r−1)/r (r is an integer equal to or greater than 2 andless than q). The present embodiment shows a configuration example ofanother encoder that creates an LDPC-CC of a time varying period of g (gis a natural number) supporting a plurality of coding rates of (r−1)/r(r is an integer equal to or greater than 2 and less than q).

FIG. 39 is a configuration example of an encoder according to thepresent embodiment. In the encoder in FIG. 39, components common tothose in FIG. 37 are assigned the same reference numerals as those inFIG. 37 and descriptions thereof will be omitted.

In encoder 800 in FIG. 37, parity computing section 830 is an encoderthat obtains parity of a coding rate of ½, ⅔, ¾ or ⅘ and paritycomputing section 840 is an encoder that obtains parity of a coding rateof ½, while in encoder 800A in FIG. 39, both parity computing section830A and parity computing section 840A perform encoding of an LDPC-CC ofa coding rate of ⅔ and a time varying period of 3, and parity computingsection 830A and parity computing section 840A are codes defined bydifferent parity check polynomials.

When coding rate setting section 810 designates coding rate ⅔, controlsection 820A performs control so that information is not inputted toparity computing section 840A. Furthermore, when coding rate ½ is set,control section 820A performs control so that the same information asthe information inputted to parity computing section 830A is inputted toparity computing section 840A.

Parity computing section 830A is an encoder that obtains parity of acoding rate of ⅔ defined, for example, by equation 45-1, equation 45-2and equation 45-3. When coding rate setting section 810 designatescoding rates ½ and ⅔, parity computing section 830A outputs parity P.

Parity computing section 840A is an encoder that obtains parity of acoding rate of ⅔ defined by a parity check polynomial different fromthat of parity computing section 830A. Parity computing section 840Aoutputs parity Pa only when coding rate setting section 810 designatescoding rate ½.

Thus, when coding rate ½ is designated, encoder 800A outputs parity Pand parity Pa for two information bits, and therefore encoder 800A canrealize coding rate ½.

It goes without saying that in FIG. 39, the coding rates of paritycomputing section 830A and parity computing section 840A are not limitedto ⅔, but the coding rates can also be ¾, ⅘, . . . , as long as paritycomputing section 830A and parity computing section 840A have the samecoding rate.

The embodiments of the present invention have been described so far. Theinvention relating to the LDPC-CC described in Embodiment 1 toEmbodiment 4 and the invention relating to the relationship between theinformation size and termination size described in the embodiments fromEmbodiment 5 onward hold true independently of each other.

Furthermore, the present invention is not limited to the above-describedembodiments, and can be implemented with various changes. For example,although cases have been mainly described above with embodiments wherethe present invention is implemented with an encoder and decoder, thepresent invention is not limited to this, and is applicable to cases ofimplementation by means of a power line communication apparatus.

It is also possible to implement the encoding method and decoding methodas software. For example, provision may be made for a program thatexecutes the above-described encoding method and communication method tobe stored in ROM (Read Only Memory) beforehand, and for this program tobe run by a CPU (Central Processing Unit).

Provision may also be made for a program that executes theabove-described encoding method and decoding method to be stored in acomputer-readable storage medium, for the program stored in the storagemedium to be recorded in RAM (Random Access Memory) of a computer, andfor the computer to be operated in accordance with that program.

It goes without saying that the present invention is not limited toradio communication, and is also useful in power line communication(PLC), visible light communication, and optical communication.

The disclosure of Japanese Patent Application No. 2009-048535, filed onMar. 2, 2009, including the specification, drawings and abstract, isincorporated herein by reference in its entirety.

INDUSTRIAL APPLICABILITY

The encoder, decoder and encoding method according to the presentinvention allow, even when performing termination, the encoder anddecoder using an LDPC-CC to prevent deterioration of error correctioncapability and avoid deterioration of information transmissionefficiency.

REFERENCE SIGNS LIST

-   100 LDPC-CC encoder-   110 Data computing section-   120, 230, 632, 830, 830A, 840, 840A Parity computing section-   130, 260 Weight control section-   140 mod 2 adder-   111-1 to 111-M, 121-1 to 121-M, 221-1 to 221-M, 231-1 to 231-M Shift    register-   112-0 to 112-M, 122-0 to 122-M, 222-0 to 222-M, 232-0 to 232-M    Weight multiplier-   200, 630, 630A, 800, 800A Encoder-   210 Information generating section-   220-1 First information computing section-   220-2 Second information computing section-   220-3 Third information computing section-   240 Adder-   250, 610, 810 Coding rate setting section-   300, 720, 900 Decoder-   310 Log likelihood ratio setting section-   320 Matrix processing computing section-   321 Storage section-   322 Row processing computing section-   323 Column processing computing section-   400, 500, 600, 600A, 700, 700A Communication apparatus-   410 Coding rate determining section-   420,640 Modulating section-   510 Receiving section-   520 Log likelihood ratio generating section-   530, 710 Control information generating suction-   620 Transmission information generation and information length    detection section-   631 Termination sequence length determining section-   633, 730 Coding rate adjusting section-   820, 820A, 910 Control section-   920, 930 BP decoding section

The invention claimed is:
 1. An encoder that performs low density paritycheck convolutional code coding, comprising: a determining section thatdetermines a sequence length of a termination sequence transmitted bybeing added at a rear end of an information sequence according to aninformation length and coding rate of the information sequence; and acomputing section that applies low density parity check convolutionalcode coding to the information sequence and a known information sequencenecessary to generate the termination sequence of the determinedsequence length, and computes a parity sequence.
 2. The encoderaccording to claim 1, further comprising an adjusting section thatchanges the number of coding rates supported according to theinformation length and adjusts the coding rate to one of the supportedcoding rates.
 3. An encoding method comprising: determining a sequencelength of a termination sequence transmitted by being added at a rearend of an information sequence according to an information length andcoding rate of the information sequence; and applying low density paritycheck convolutional code coding to the information sequence and a knowninformation sequence necessary to generate the termination sequence ofthe determined sequence length and computing a parity sequence.